\begin{proposition}\label{prop: rank 3 excluded minor}
    If $M$ is a simple rank-$3$ matroid with a line $L$ and distinct points $e_1, e_2, e_3$ on $L$ each on at least two long lines, then $M$ is not a positroid.
    In particular, $M$ has a minor from Figure \ref{fig: rank-3 excluded minors}.
\end{proposition}

\begin{theorem} \label{thm: main}
For all integers $r, \ell \ge 1$, if $M$ is a simple rank-$r$ positroid with no $U_{2,\ell+2}$-minor, then $|M| \le \ell(r - 1) + 1$.
Moreover, if $r \ge 2$ then equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2, \ell+1}$.
\end{theorem}

\begin{proposition}\label{prop: rank 4 excluded minor}
    If $M$ is a simple rank-$4$ matroid with a line $L$, distinct points $e_1, e_2, e_3$ on $L$, and distinct planes $P_1$, $P_2$, $P_3$ through $L$ so that for $i = 1,2,3$ the point $e_i$ is on at least two long lines in $P_i$, then $M$ is not a positroid.
    In particular, $M$ has a minor from Figure \ref{fig: rank-3 excluded minors} or Figure \ref{fig: rank-4 excluded minor}.
\end{proposition}

\label{sec: introduction}

Extremal matroid theory was formally introduced in 1993 with the following problem posed by Joseph Kung \cite{Kung-1993}.

\begin{problem} \label{prob: Kung problem}
Let $\cM$ be a class of matroids.
Determine the maximum number of elements of a simple rank-$r$ matroid in $\cM$, and characterize the matroids for which equality holds.
\end{problem}

This maximum does not exist for all classes of matroids.
For instance, there is no finite answer to Problem~\ref{prob: Kung problem} for the class of all matroids, or for any class containing all uniform matroids.
However, Kung \cite[Theorem 4.3]{Kung-1993} proved that every simple rank-$r$ matroid with no $U_{2, \ell+2}$-minor has at most $\frac{\ell^r - 1}{\ell - 1}$ elements.
If $\ell$ is a prime power, then this bound is tight and is achieved by the rank-$r$ projective geometry over the finite field of order $\ell$. It is thus natural to study Problem~\ref{prob: Kung problem} for classes of matroids without $U_{2,\ell}$-minors.

In this paper, we study the class of \textit{positroids} without $U_{2,\ell}$-minors. Positroids, which are matroids representable by a matrix over $\bR$ with all maximal subdeterminants nonnegative, occupy a central position at the intersection of combinatorics, algebraic geometry, and physics. They serve as combinatorial models for the totally nonnegative Grassmannian, which in turn underlies the construction of the amplituhedron in quantum field theory. The study of amplituhedra and related objects remains an active and influential area in theoretical physics \cite{AHT}. Through these connections, positroids emerge as unifying objects that integrate discrete combinatorial frameworks with geometric and physical theory. Combinatorially, they are remarkably rich: Postnikov~\cite{postnikov} showed that positroids are in bijection with decorated permutations, Grassmann necklaces, plabic graphs, and \reflectbox{L}-diagrams, providing multiple equivalent viewpoints.
Positroids also form a minor-closed class of matroids. It is thus natural to ask extremal questions about positroids and their minors. There has been work enumerating both positroids \cite{Williams05} and connected positroids \cite{ARW16}, but as far as the authors are aware, no other extremal results about positroids are known. Our main result tightly resolves Problem~\ref{prob: Kung problem} for the class of $U_{2,\ell+2}$-minor-free positroids.

This result lies at the intersection of three areas of research: extremal matroid theory, structural properties of positroids, and structural properties of matroids with no $M(K_4)$-minor.
We will explain how Theorem \ref{thm: main} continues lines of research in these three areas, beginning with extremal matroid theory.
The following beautiful result of Geelen and Nelson \cite[Theorem 1.2]{Geelen-Nelson-2010} generalizes the initial work of Kung and tightly solves Problem \ref{prob: Kung problem} for the class of matroids with no $U_{2,\ell+2}$-minor, for all values of $\ell$ and all sufficiently large values of $r$.

\begin{theorem}[Geelen, Nelson \cite{Geelen-Nelson-2010}] \label{thm: exlude a line}
Let $\ell\ge 2$ be an integer and let $q$ be the largest prime power at most $\ell$.
For $r$ sufficiently large, every simple rank-$r$ matroid with no $U_{2,\ell+2}$-minor has at most $\frac{q^r - 1}{q - 1}$ elements, with equality only for the rank-$r$ projective geometry over the order-$q$ finite field.
\end{theorem}

\begin{theorem}[Geelen, Nelson, Walsh \cite{Geelen-Nelson-Walsh-2024}] \label{thm: complex exclude a line}
Let $\ell\ge 2$ be an integer and let $r$ be a sufficiently large integer.
Then every simple rank-$r$ $\bC$-representable matroid with no $U_{2,\ell+2}$-minor has at most $\ell\binom{r}{2} + r$ elements, with equality only for the rank-$r$ Dowling geometry over the cyclic group of order $\ell$.
\end{theorem}

\begin{theorem} \label{thm: main result with proof}
For all integers $r, \ell \ge 1$, if $M$ is a simple rank-$r$ matroid with no minor from Figure \ref{fig: rank-3 excluded minors} or Figure \ref{fig: rank-4 excluded minor} and no $U_{2,\ell+2}$-minor, then $|M| \le \ell(r - 1) + 1$.
Moreover, equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2, \ell+1}$.
\end{theorem}
\begin{proof}
First suppose that $\ell = 1$.
If $M$ is a simple rank-$r$ matroid with more than $r$ elements, then $r \ge 2$ and $M$ has a circuit with at least three elements.
But every circuit with at least three elements has a $U_{2,3}$-minor \cite[Example 3.1.9]{Oxley-2011}, a contradiction.
So $M$ is a rank-$r$ matroid with $r$ elements, and it follows that $M$ is the parallel connection of $r - 1$ copies of $U_{2,2}$, as desired.
So we may assume that $\ell \ge 2$.
We will proceed by induction on $r$.
The statement is clearly true when $r \le 2$ so we may assume that $r \ge 3$.
If $M$ has an element $e$ that is not on any long lines, then $M/e$ is simple, $|M/e| = |M| - 1$, and $r(M/e) = r - 1$, so by induction we have
\begin{align*}
 |M| = 1 + |M/e| \le 1 + \ell(r - 2) + 1 = \ell(r - 1) + 2 - \ell < \ell(r - 1) + 1,
\end{align*}
as desired.
So we may assume that every element of $M$ is on a long line.
Let $L$ be a long line of $M$.
We will now consider the connected components of $M/L$.
Let $\cX$ be the partition of $E(M) - L$ given by the ground sets of the connected components of $M/L$.
Let $X \in \cX$, and let $N = M|(L \cup X)$.
We will first bound the number of elements of $N \del L$.

\begin{claim} \label{claim: bound the size of each component}
 At most two elements in $L$ are on a long line of $N$ other than $L$, and $|N \del L| \le \ell(r(N) - 2)$.
\end{claim}
\begin{proof}
Suppose that distinct elements $e_1$, $e_2$, and $e_3$ in $L$ are on long lines $L_1$, $L_2$, and $L_3$, respectively, of $N$ other than $L$.
Let $K = \si(N/L)$.
By deleting elements from nontrivial parallel classes of $N/L$ we may assume that $K$ is a restriction of $N/L$.
Note that the elements of $K$ are in one-to-one correspondence with the planes of $N$ containing $L$.
For $i = 1,2,3$ let $p_i$ be the element of $K$ in the plane $\cl_N(L \cup L_i)$ of $N$.
Since $K$ is connected, it follows from \cite[Theorem 4.3.1]{Oxley-2011} that $K$ has a connected minor $K_1 = K \del D/C$ on ground set $\{p_1, p_2, p_3\}$.
We may assume that $C$ is independent in $K$ and therefore in $N/L$, which implies that $r_{N/C}(L) = r_N(L) = 2$.
Since $K_1$ is connected, it is isomorphic to $U_{1,3}$ or $U_{2,3}$.
Let $N_1$ be obtained from $N$ by deleting, for each element $d$ in $D$, the elements in $\cl_N(L \cup d) - L$, and then contracting $C$.
Then $K_1 = \si(N_1/L)$ and $r(N_1) = r(K_1) + 2$.
We will show that $N_1$ has a minor from Figure \ref{fig: rank-3 excluded minors} or \ref{fig: rank-4 excluded minor}.

\begin{claim}
 Some $x \in L$ is on more than one long line of $M$.
\end{claim}
\begin{proof}
Let $e \in L$ and let $M_1 = M/e$.
Let $P = L - e$, and note that $P$ is a nontrivial parallel class of $M_1$.
Also note that $\elem(M_1) = |M| - \ell = \ell(r(M_1) - 1) + 1$,
so by induction, $\si(M_1)$ is obtained by taking parallel connections of copies of $U_{2,\ell+1}$.
This implies that $P$ is on a long line $L_1$ of $M_1$.
Let $F$ be the plane $L \cup L_1$ of $M$.
Note that $|F| \ge |L| + |L_1| - 1 \ge 2(\ell + 1) - 1 = 2\ell + 1$.
If $F - L$ has a $3$-element independent set $I$, then $M|F$ has an $M(K_4)$-restriction (if each pair of points in $I$ spans a point on $L$) or a $U_{2, \ell+2}$-minor, a contradiction.
So $F - L$ has rank $2$.
If $\cl(F - L)$ does not intersect $L$, then for every $f \in F - L$ we have $(M|F)/f \cong U_{2,\ell+2}$, a contradiction.
So $\cl(F - L)$ intersects $L$ in some element $x$.
Since $|F - L| \ge \ell \ge 2$, it follows that $\cl(F - L)$ is a long line of $M$ that contains $x$ and is not equal to $L$, as desired.
\end{proof}

Let $r, \ell$ be integers with $r \ge 2$ and $\ell \ge 3$.
Let $W(r,\ell)$ be the matroid with a basis $B = \{b_1, b_2, \dots, b_r\}$ and $\lfloor \frac{\ell - 1}{2} \rfloor$ elements freely placed (see \cite[page 270]{Oxley-2011}) in the span of $\{b_i, b_{i+1}\}$ for all $i \in [r]$, reading indices modulo $r$.
Let $W(r, \ell)^+$ be obtained from $W(r, \ell)$ by freely placing one point in $\cl(\{b_1, b_2\})$.
Then $W(2, \ell) \cong U_{2,\lfloor \frac{\ell - 1}{2} \rfloor + 2}$, and $W(r, 3)$ and $W(r, 4)$ are both isomorphic to the rank-$r$ whirl $\mathcal W^r$ when $r \ge 3$.
Since the whirl $\mathcal W^r$ is $3$-connected (see \cite[Example 8.3.4]{Oxley-2011}) and $W(r,\ell)$ has $\mathcal W^r$ as a restriction, it follows from \cite[page 295, Exercise 5]{Oxley-2011} that $W(r,\ell)$ and $W(r, \ell)^+$ are $3$-connected.
And it is straightforward to check that the natural cyclic ordering of $E(W(r, \ell))$ (first $b_1$, then all elements in $\cl(\{b_1, b_2\}) - b_2$, then $b_2$, and so on) or $W(r,\ell)^+$ satisfies the condition of Theorem \ref{thm:Bonin's condition}, so $W(r, \ell)$ and $W(r, \ell)^+$ are positroids.
Finally, one can check that if $r \ge 4$ and $e \notin \{b_1, \dots, b_r\}$, then $\si(W(r, \ell)/e) \cong W(r, \ell)$, and that $\si(W(r, \ell)/b_i) \cong W(r,\ell) \del (\cl(\{b_{i-1},b_i, b_{i+1}) - \{b_{i-1}, b_{i+1}\})$, and from these observations one can prove that $W(r,\ell)$ is $U_{2,\ell+2}$-minor-free, and when $\ell$ is even, $W(r, \ell)^+$ is $U_{2, \ell+2}$-minor-free.
Noting that $|W(r, \ell)| = r + r\lfloor \frac{\ell-1}{2} \rfloor$, we make the following conjecture.

This result lies at the intersection of three areas of research: extremal matroid theory, structural properties of positroids, and structural properties of matroids with no $M(K_4)$-minor.
We will explain how Theorem \ref{thm: main} continues lines of research in these three areas, beginning with extremal matroid theory.
The following beautiful result of Geelen and Nelson \cite[Theorem 1.2]{Geelen-Nelson-2010} generalizes the initial work of Kung and tightly solves Problem \ref{prob: Kung problem} for the class of matroids with no $U_{2,\ell+2}$-minor, for all values of $\ell$ and all sufficiently large values of $r$.

\begin{theorem}[Geelen, Nelson \cite{Geelen-Nelson-2010}] \label{thm: exlude a line}
Let $\ell\ge 2$ be an integer and let $q$ be the largest prime power at most $\ell$.
For $r$ sufficiently large, every simple rank-$r$ matroid with no $U_{2,\ell+2}$-minor has at most $\frac{q^r - 1}{q - 1}$ elements, with equality only for the rank-$r$ projective geometry over the order-$q$ finite field.
\end{theorem}

Projective geometries are not representable over fields of characteristic zero (see \cite[page 205, Exercise 8]{Oxley-2011}), inviting the following question: for a field $\bF$ with characteristic zero, what is the maximum number of elements of a simple rank-$r$ $\bF$-representable matroid with no $U_{2, \ell+2}$-minor?
This was answered by Geelen, Nelson, and Walsh \cite[Theorem 1.0.5]{Geelen-Nelson-Walsh-2024} in the case that $\bF = \bC$.

\begin{theorem}[Geelen, Nelson, Walsh \cite{Geelen-Nelson-Walsh-2024}] \label{thm: complex exclude a line}
Let $\ell\ge 2$ be an integer and let $r$ be a sufficiently large integer.
Then every simple rank-$r$ $\bC$-representable matroid with no $U_{2,\ell+2}$-minor has at most $\ell\binom{r}{2} + r$ elements, with equality only for the rank-$r$ Dowling geometry over the cyclic group of order $\ell$.
\end{theorem}

When $\bF \in \{\bR, \mathbb Q\}$, there is a lower bound of $2\binom{r}{2} + r$ and an upper bound of $2 \binom{r}{2} + f(\ell)\cdot r$ with $f$ a function of $\ell$, due to Geelen, Nelson, and Walsh \cite[Theorem 1.0.6]{Geelen-Nelson-Walsh-2024}, but no exact solution to Problem~\ref{prob: Kung problem} is known for all $\ell$.
Theorem \ref{thm: main} can be seen as a further specialization of Theorems \ref{thm: exlude a line} and \ref{thm: complex exclude a line} to the subclass of $\bR$-representable matroids consisting of positroids.

Theorem \ref{thm: main} also continues a recent line of research that studies positroids from a structural perspective.
For example, Bonin \cite{Bonin-2024} studied matroid amalagams of positroids, Quail \cite{Quail-2024} studied positroids representable over the finite fields of order $3$ and of order $4$, Quail and Rombach \cite{Quail-Rombach-2025} studied graphic positroids, and Park \cite{Park-2023} studied excluded minors for positroids.
Our proof of Theorem \ref{thm: main} relies on two new structural properties (Propositions \ref{prop: rank 3 excluded minor} and \ref{prop: rank 4 excluded minor}) for low-rank positroids.

Finally, \Cref{thm: main} contributes to a line of research studying matroids with no $M(K_4)$-minor, where $M(K_4)$ is the graphic matroid of the complete graph $K_4$.
For example, Sims \cite{Sims-1977} proved that the class of $M(K_4)$-minor-free matroids is complete, Brylawski \cite{Brylawski-1971} characterized the binary matroids with no $M(K_4)$-minor, and Oxley \cite{Oxley-1987} characterized the $3$-connected ternary matroids with no $M(K_4)$-minor.
More recently, work of Pendavingh and van der Pol \cite{Pendavingh-vanderPol-2015} and van der Pol \cite{vanderPol-2023} investigated the importance of $M(K_4)$ in matroid enumeration.
Most notably for this paper, Kung studied Problem \ref{prob: Kung problem} for the class of matroids with no $M(K_4)$-minor and no $U_{2, \ell+2}$-minor, proving that every such simple rank-$r$ matroid has at most $(6\ell^{\ell - 1} + 8\ell)\cdot r$ elements \cite[Theorem 1.1]{Kung-1988}. The graphic matroid $M(K_4)$ is not a positroid, so this also bounds from above the maximum number of elements of a simple rank-$r$ positroid with no $U_{2, \ell+2}$-minor.
To the best of our knowledge, this was in fact the previous best upper bound for this quantity.
There are several other notable $M(K_4)$-minor-free classes of matroids that contain the class of positroids, such as gammoids, strongly base-orderable matroids, and $k$-base-orderable matroids.
We will discuss these further in Section \ref{sec: future work}.

The rest of this paper is structured as follows.
In Section \ref{sec: preliminaries}, we will give more background on matroids and positroids.
In Section \ref{sec: excluded minors}, we will use a list of $9$ known excluded minors for the class of positroids to prove two structural properties for low-rank positroids.
In Section \ref{sec: the extremal examples}, we will describe the matroids for which equality holds in Theorem \ref{thm: main} and prove that they are in fact positroids and are $U_{2, \ell+2}$-minor-free.
We will prove Theorem \ref{thm: main} in Section \ref{sec: the main proof}, and conclude in Section \ref{sec: future work} by discussing several directions for future work.

Our proof of Theorem \ref{thm: main} relies on two new structural properties (Propositions \ref{prop: rank 3 excluded minor} and \ref{prop: rank 4 excluded minor}) for low-rank positroids. The paper then (i) uses a list of $9$ known excluded minors for the class of positroids to prove these two structural properties for low-rank positroids (Section \ref{sec: excluded minors}); (ii) describes the matroids for which equality holds in Theorem \ref{thm: main} and proves that they are positroids and are $U_{2, \ell+2}$-minor-free (Section \ref{sec: the extremal examples}); and (iii) proves Theorem \ref{thm: main} in Section \ref{sec: the main proof}.

Inequality or Bound
Biconditional or Equivalence

Question: Let $r,2 ell\ge 1$ be integers. Recall that a positroid is a matroid representable over $\mathbb{R}$ by a matrix whose maximal minors are all nonnegative, and that $U_{2,m}$ denotes the rank-$2$ uniform matroid on $m$ elements. If $M$ is a simple rank-$r$ positroid with no $U_{2,\ell+2}$-minor, which quantitative estimate for the number of elements $|M|$ holds, and for $r\ge 2$ when does equality occur?

The bound $|M|\le \ell(r-1)+1$ always holds. Moreover, if $r\ge 2$, then equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2,\ell+1}$.

Choices:
B. The bound $|M|\le \ell r+1$ always holds. Moreover, if $r\ge 2$, then equality holds if and only if $M\cong U_{2,\ell+2}$ or, more generally, $M$ is obtained by taking parallel connections of copies of $U_{2,\ell+2}$.
C. The bound $|M|\le \ell(r-1)+1$ always holds.
D. For each fixed $\ell\ge 1$, there exists an integer $r_0(\ell)$ such that whenever $r\ge r_0(\ell)$ and $M$ is a simple rank-$r$ positroid with no $U_{2,\ell+2}$-minor, one has $|M|\le \ell(r-1)+1$; moreover, for such sufficiently large $r$, equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2,\ell+1}$.
E. The bound $|M|\le \ell(r-1)+1$ always holds. Moreover, if $r\ge 2$, then equality holds if and only if every long line of $M$ has exactly $\ell+1$ points.

all-ranks statement replaced by sufficiently-large-r

global parallel-connection structure replaced by local long-line condition

The stem does not explicitly reveal the equality characterization, and it does not mention parallel connections or the specific building block $U_{2,\ell+1}$. It states the bound and asks for the equivalent equality case without giving away the correct option.

This is close to theorem recall: the stem presents the extremal bound and asks for the equality characterization. However, it is not a verbatim restatement because the student must still distinguish among several structurally similar conclusions.

Some reasoning is required to reject plausible alternatives such as a direct sum, a weaker minor statement, or the wrong line size. Still, the item mainly tests recognition of the exact characterization rather than substantial derivation.

The distractors are plausible and well-targeted: B confuses parallel connection with direct sum, C is a weaker true-type statement, D overgeneralizes the line sizes, and E incorrectly uses the excluded threshold. These reflect realistic mathematical failure modes.

A solid MCQ with no real answer leakage and strong distractors, but it leans heavily on theorem recall and is only moderately generative rather than deeply reasoning-based.


\begin{theorem}\label{main} Let $T>0$, $l_j>0,\ j=1,...,N$, $s\in [0,3]$ be given. Suppose $f\in W^{\frac{s}{3},1}(0,T;\mathbb{L}^2(\mathcal{T}))\cap L^{\frac{6}{6-s}}(0,T;\mathbb{H}^\frac{s}{3}(\mathcal{T}))$ and
\begin{align*}
(u^0,g_0,g,p)\in \mathbb{H}^s(\mathcal{T})\times H^{\frac{s-1}{3}}(0,T)\times \left[H^\frac{s}{3}(0,T)\right]^{N}\times \left[H^{\frac{s+1}{3}}(0,T)\right]^N.
\end{align*}
satisfying the $s$-compatibility conditions, thus there exists $T^*\in (0,T)$ such that the IBVP \eqref{kdV} admits a unique solution $u \in \mathcal{X}_{s,T^*}$, where
$$\mathcal{X}_{s,T}=\left\{v \in \mathbb{B}^{*}_{s,T};\ \partial_x^ku_j\in L_x^\infty(0,l_j;H^\frac{s+1-k}{3}(0,T)),\ k=0,1,2,\ j=1,2,...,N\right\},$$
\begin{align*}
	\mathbb{B}_{s,T}^*:=\mathbb{B}_{s,T}\cap H^\frac{s}{3}(0,T;\mathbb{H}^1(\mathcal{T})).
\end{align*}
\end{theorem}

In mathematics and physics, a quantum graph is a network composed of vertices connected by edges, where each edge carries a differential or pseudo-differential equation. When every edge is endowed with a natural metric, the structure is referred to as a metric graph. A common illustration is a power network: the transmission lines act as edges connecting transformer stations (vertices). In such a case, the governing differential equations may describe the voltage along each line, with boundary conditions at the vertices ensuring that the total current flowing into and out of each vertex sums to zero.

The concept of quantum graphs traces back to the work of Linus Pauling in the 1930s, who used them to model free electrons in organic molecules. Since then, they have emerged in a wide range of contexts, including quantum chaos, waveguide analysis, photonic crystals, and Anderson localization—the suppression of wave propagation in disordered media. Quantum graphs also arise as limit models of systems composed of thin wires and serve as important theoretical tools in mesoscopic physics and nanotechnology. A simplified formulation of quantum graphs was later proposed by Freedman \textit{et al.} in \cite{Freedman}.

Beyond solving the differential equations in specific applications, the study of quantum graphs encompasses fundamental questions such as well-posedness, controllability, and identifiability. For example, controllability concerns determining the inputs required to steer the system toward a desired configuration, such as ensuring stable power distribution across a grid. Identifiability, on the other hand, focuses on optimal measurement placement—for instance, choosing pressure gauges in a water network to detect and locate potential leaks.

\subsection{Dispersive systems on graph structure}
In recent years, the study of nonlinear dispersive equations on metric graphs has attracted growing interest from mathematicians, physicists, chemists, and engineers (see \cite{BK, BlaExn08, BurCas01, K, Mug15} for detailed discussions and references). A fundamental setting for such investigations is the star graph $\mathcal{G}$, a metric graph consisting of $N$ half-lines $(0,\infty)$ meeting at a single common vertex $\nu = 0$. On each edge, one typically defines a nonlinear dispersive equation—most prominently, the nonlinear Schrödinger equation (see the works of Adami \textit{et al.} \cite{AdaNoj14, AdaNoj15} and Angulo and Goloshchapova \cite{AngGol17a, AngGol17b}).
The introduction of nonlinearities into dispersive models on such networks creates a rich framework for exploring soliton propagation, energy transfer, and nonlinear wave interactions. A central analytical difficulty arises at the vertex, where the graph may exhibit bifurcation or multi-bifurcation phenomena, especially in more intricate network topologies.

Other nonlinear dispersive systems on graphs have also been the subject of extensive study. For example, in the context of well-posedness theory, Cavalcante \cite{Cav} established local well-posedness for the Cauchy problem associated with the Korteweg–de Vries (KdV) equation on a metric star graph composed of one negative and two positive half-lines joined at a common vertex $\nu = 0$ (the so-called $\mathcal{Y}$-junction).

Another important example is the Benjamin–Bona–Mahony (BBM) equation: Bona and Cascaval \cite{bona} proved local well-posedness in the Sobolev space $H^1$, while Mugnolo and Rault \cite{Mugnolobbm} showed the existence of traveling-wave solutions on graphs. Using a different approach, Ammari and Crépeau \cite{AmCr1} derived results concerning well-posedness and stabilization of the BBM equation in star-shaped networks with bounded edges. Moreover, Mugnolo \textit{et al.} \cite{MugNoSe} provided a full characterization of boundary conditions under which the Airy-type evolution equation
$$u_t = \alpha u_{xxx} + \beta u_x, \quad \alpha \in \mathbb{R} \setminus \{0\}, \ \beta \in \mathbb{R},$$
generates contraction semigroups on star graphs. We also refer to the recent work on Airy dynamics on graphs with looping edges \cite{PavaMunoz24}, which provides a complementary perspective on this setting.
\medskip

Significant progress has also been achieved in the fields of control theory and inverse problems on networks. Ignat \textit{et al.} \cite{Ignat2011} examined inverse problems for the heat and Schrödinger equations on tree-like structures, while Baudouin and Yamamoto \cite{Baudouin} proposed a unified and simplified framework for coefficient identification problems. Furthermore, Ammari and Crépeau \cite{Ammari and Crepeau 2018}, Cerpa \textit{et al.} \cite{Cerpa, Cerpa1}, and Parada \textit{et al.} \cite{Parada2022a, Parada2022b} established stabilization and boundary controllability results for the KdV equation on star-shaped graphs, contributing substantially to the control-theoretic understanding of dispersive models on networks. Recently, in \cite{CapistranoParadaDaSilva25}, we studied the analytical structure of the entire functions associated with the KdV operator, addressing control problems involving mixed Dirichlet–Neumann boundary controls.

\subsection{Setting of problem and functional framework} In the context of graph structure, let us consider the following: given $N\geq 2$, consider a star-shaped network $\mathcal{T}$ with $N$ edges described by intervals $I_j=(0,l_j)$, $l_j>0$ for $j=1,...,N$. Denoting by $e_1,...,e_N$ the edges of $\mathcal{T}$ one has $\mathcal{T}=\bigcup_{j=1}^Ne_j$. Here, we study the local  well-posedness for a system composed of $N$ nonlinear KdV equations posed on $\mathcal{T}$ and coupled by the boundary, namely
		\begin{align}\label{kdV}
			\begin{cases}
				\partial_t u_j+\partial_x u_j+\partial_x^3 u_j+u_j\partial_x u_j=0,&t \in (0,T),\ x \in (0,l_j),\ j=1,...,N\\
				u_j(t,0)=u_1(t,0),&t\in (0,T),\ j=1,...,N\\
				\displaystyle\sum_{j=1}^N\partial_x^2 u_j(t,0)=-\alpha u_1(t,0)-\frac{N}{3}u_1^2(t,0)+g_0(t),&t \in (0,T)\\
				u_j(t,l_j)=p_j(t),\ \ \ \ \ \partial_x u_j(t,l_j)=g_j(t),&t\in (0,T),\ j=1,...,N\\
				u_j(0,x)=u_j^0(x),&x \in (0,l_j),\ j=1,...,N
			\end{cases}
		\end{align}
where $u=(u_1,...,u_N)$ stands the state of the system, $p_j$ and $g_j$ are the controls inputs and $\alpha>\frac{N}{2}$. We represent this situation in the figure \eqref{fig:network}.
		\begin{figure}[H]
			\centering
			\begin{tikzpicture}
				\draw(0, 0) circle (1mm);
				\draw[line width=0.8pt](0.02, 0.09) -- (0.5, 1.95);
				\draw[fill=black](0.53, 2.04) circle (1mm);
				\draw[line width=0.8pt](0.07, 0.07) -- (1.33, 1.5);
				\draw[fill=black](1.39, 1.57) circle (1mm);
				\draw[line width=0.8pt](0.1, 0.04) -- (1.85, 0.78);
				\draw[fill=black](1.93, 0.815) circle (1mm);
				\draw[line width=0.8pt](0.08, -0.04) -- (1.75, -1);
				\draw[fill=black](1.825, -1.045) circle (1mm);
				\draw[line width=0.8pt](0.05, -0.09) -- (1.02, -1.73);
				\draw[fill=black](1.07, -1.8) circle (1mm);
				\draw[fill=black](1.45, 0.2) circle (0.3mm);
				\draw[fill=black](1.45, 0.05) circle (0.3mm);
				\draw[fill=black](1.45, -0.1) circle (0.3mm);
				\draw node (no1) at (-0.3,0) {$0$};
				\draw node (no1) at (0.19, 1.7) {$e_1$};
				\draw node (no1) at (0.86, 1.35) {$e_2$};
				\draw node (no1) at (1.35, 0.8) {$e_3$};
				\draw node (no1) at (1.7, -0.6) {$e_{_{N-1}}$};
				\draw node (no1) at (1.06, -1.25) {$e_{_N}$};
				\draw(3.95,-0.5) circle (1mm);
				\draw node (no1) at (6.4,-0.5) {{\footnotesize \hspace{10mm} internal controlled node}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ };
				\draw[fill=black](3.95,-1) circle (1mm);
				\draw node (no1) at (6.4,-1) {{\footnotesize \hspace{13mm} external controlled node}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ };
			\end{tikzpicture}
			\label{fig:network}
			\caption{Network with $N$ edges}
	\end{figure}

Let us introduce some notation used throughout this work. From now on, we have the following:
	 \begin{itemize}
	 \item[i.] The vector $u$ is given by $u=(u_1,...,u_N) \in \mathbb{L}^2(\mathcal{T})=\displaystyle\prod_{j=1}^NL^2(0,l_j)$
	and initial data is $u^0=(u^0_1,...,u^0_N)$.
	 \item[ii.] The inner product in $\mathbb{L}^2(\mathcal{T})$ will be given by  $$\left(u,z\right)_{\mathbb{L}^2(\mathcal{T})}=\sum_{j=1}^N\int_0^{l_j}u_jz_jdx,\ \ u,z\in \mathbb{L}^2(\mathcal{T}).$$
	Moreover,  $$\mathbb{H}^s(\mathcal{T})=\displaystyle\prod_{j=1}^NH^s(0,l_j), \ \ s \in \mathbb{R}.$$
	 \end{itemize}
	\begin{itemize}
		\item[iii.] Define also the following spaces:
		$$\mathbb{H}_0^k(\mathcal{T})=\displaystyle\prod_{j=1}^{N}H_0^k(0,l_j),\ k\in \mathbb{N}$$
		and
		$$H_r^k(0,l_j)=\left\{v \in H^k(0,l_j),\ \partial_x^iv(l_j)=0,\ 0\leq i\leq k-1\right\},\ k=1,...,N,$$
		where $v^{(m)}=\frac{dv}{dx^m}$ and the index $r$ is related to the null right boundary conditions. In addition,
		$$\mathbb{H}^k_r(\mathcal{T})=\prod_{j=1}^{N}H_r^k(0,l_j)\quad \text{and} \quad \|u\|_{\mathbb{H}^k_r(\mathcal{T})}^2=\sum_{j=1}^{n}\|u_j\|_{H^k(0,l_j)}^2\ k \in \mathbb{N}.$$
		\item[iv.] Consider the following characterization:
		$$H_r^{-1}(0,l_j)=\left(H_r^{1}(0,l_j)\right)'$$
		as the dual space of $H_r(0,l_j)$ with respect to the pivot space $L^2(0,l_j)$ and $\mathbb{H}_r^{-1}$ denotes the cartesian product of $H_r^{-1}(0,l_j)$.
\item[v.] For $k\in \mathbb{N}$, the derivative $\partial_x^ku$ means the row vector
\begin{align*}
\partial_x^k u:=(\partial_x^ku_1,...,\partial_x^ku_N).
\end{align*}
The derivative $\partial_t^ku$ is understood in the same way.
\item[vi.] Under the above notation, we have
\begin{align*}
	C^k\left([0,T];\mathbb{H}^s(\mathcal{T})\right)=\prod_{j=1}^NC^k\left([0,T];H^s(0,l_j)\right),\ k=0,1,2,... \text{ \ and \ }s \in \mathbb{R}
\end{align*}
as well as
\begin{align*}
	L^p\left(0,T;\mathbb{H}^s(\mathcal{T})\right)=\prod_{j=1}^NL^p\left(0,T;H^s(0,l_j)\right),\ p\geq 1 \text{ \ and \ }s \in \mathbb{R}.
\end{align*}
Hence, for $f \in 	C^k\left([0,T];\mathbb{H}^s(\mathcal{T})\right)$ and $q \in L^p\left(0,T;\mathbb{H}^s(\mathcal{T})\right)$ we can write
\begin{align*}
f=(f_1,...,f_N)\text{ \ and \ }q=(q_1,...,q_N)
\end{align*}
with
\begin{align*}
f_j \in C^k\left([0,T];H^s(0,l_j)\right)\text{ \ and \ }q_j \in L^p\left(0,T;H^s(0,l_j)\right),\ j=1,...N.
\end{align*}
\item[vii.] We will consider the spaces
$$\mathbb{H}^k_e(\mathcal{T})=\left\{u=(u_1,...,u_N)\in \mathbb{H}^k_r(\mathcal{T});\ u_j(0)=u_k(0),\ j,k=1,...,N\right\}\ k \in \mathbb{N},$$
$$\mathbb{H}^3_c(\mathcal{T})=\left\{u=(u_1,...,u_N)\in \mathbb{H}^3(\mathcal{T});\ u_j(0)=u_k(0),\ j,k=1,...,N\right\},$$
and
$$\mathbb{B}_{s,T}=C([0,T],\mathbb{H}^s(\mathcal{T}))\cap L^2(0,T,\mathbb{H}^{s+1}(\mathcal{T})),$$
with the norm
$$\displaystyle\|u\|_{\mathbb{B}_{s,T}}:=\|u\|_{C([0,T],\mathbb{H}^s(\mathcal{T}))}+\|u\|_{L^2(0,T,\mathbb{H}^{s+1}(\mathcal{T}))}=\max_{t\in [0,T]}\|u\|_{\mathbb{H}^s(\mathcal{T})}+\left(\int_0^T\|u(t,\cdot)\|_{\mathbb{H}^{s+1}}^2dt\right)^\frac{1}{2}.$$
When $s=0$ we write $\mathbb{B}_T$ instead $\mathbb{B}_{0,T}$.
\item [viii.] Once understand the appropriate spaces in each situation, we will use the notations $g=(g_1,...,g_N)$ and $p=(p_1,...,p_N)$.
\end{itemize}

\subsection{Main results and structure of the article} With the previous framework in hand, our goal here is to prove the local and global well-posedness for the initial boundary value problem (IBVP) \eqref{kdV}. The first main result establishes the local well-posedness for this system in sharp spaces.

\subsection{Dispersive systems on graph structure}
In recent years, the study of nonlinear dispersive equations on metric graphs has attracted growing interest from mathematicians, physicists, chemists, and engineers (see \cite{BK, BlaExn08, BurCas01, K, Mug15} for detailed discussions and references). A fundamental setting for such investigations is the star graph $\mathcal{G}$, a metric graph consisting of $N$ half-lines $(0,\infty)$ meeting at a single common vertex $\nu = 0$. On each edge, one typically defines a nonlinear dispersive equation—most prominently, the nonlinear Schrödinger equation (see the works of Adami \textit{et al.} \cite{AdaNoj14, AdaNoj15} and Angulo and Goloshchapova \cite{AngGol17a, AngGol17b}).
The introduction of nonlinearities into dispersive models on such networks creates a rich framework for exploring soliton propagation, energy transfer, and nonlinear wave interactions. A central analytical difficulty arises at the vertex, where the graph may exhibit bifurcation or multi-bifurcation phenomena, especially in more intricate network topologies.

Another important example is the Benjamin–Bona–Mahony (BBM) equation: Bona and Cascaval \cite{bona} proved local well-posedness in the Sobolev space $H^1$, while Mugnolo and Rault \cite{Mugnolobbm} showed the existence of traveling-wave solutions on graphs. Using a different approach, Ammari and Crépeau \cite{AmCr1} derived results concerning well-posedness and stabilization of the BBM equation in star-shaped networks with bounded edges. Moreover, Mugnolo \textit{et al.} \cite{MugNoSe} provided a full characterization of boundary conditions under which the Airy-type evolution equation
$$u_t = \alpha u_{xxx} + \beta u_x, \quad \alpha \in \mathbb{R} \setminus \{0\}, \ \beta \in \mathbb{R},$$
generates contraction semigroups on star graphs. We also refer to the recent work on Airy dynamics on graphs with looping edges \cite{PavaMunoz24}, which provides a complementary perspective on this setting.
\medskip

Let us introduce some notation used throughout this work. From now on, we have the following:
 \begin{itemize}
 \item[i.] The vector $u$ is given by $u=(u_1,...,u_N) \in \mathbb{L}^2(\mathcal{T})=\displaystyle\prod_{j=1}^NL^2(0,l_j)$
 and initial data is $u^0=(u^0_1,...,u^0_N)$.
 \item[ii.] The inner product in $\mathbb{L}^2(\mathcal{T})$ will be given by $$\left(u,z\right)_{\mathbb{L}^2(\mathcal{T})}=\sum_{j=1}^N\int_0^{l_j}u_jz_jdx,\ \ u,z\in \mathbb{L}^2(\mathcal{T}).$$
 Moreover, $$\mathbb{H}^s(\mathcal{T})=\displaystyle\prod_{j=1}^NH^s(0,l_j), \ \ s \in \mathbb{R}.$$
 \end{itemize}
 \begin{itemize}
 \item[iii.] Define also the following spaces:
 $$\mathbb{H}_0^k(\mathcal{T})=\displaystyle\prod_{j=1}^{N}H_0^k(0,l_j),\ k\in \mathbb{N}$$
 and
 $$H_r^k(0,l_j)=\left\{v \in H^k(0,l_j),\ \partial_x^iv(l_j)=0,\ 0\leq i\leq k-1\right\},\ k=1,...,N,$$
 where $v^{(m)}=\frac{dv}{dx^m}$ and the index $r$ is related to the null right boundary conditions. In addition,
 $$\mathbb{H}^k_r(\mathcal{T})=\prod_{j=1}^{N}H_r^k(0,l_j)\quad \text{and} \quad \|u\|_{\mathbb{H}^k_r(\mathcal{T})}^2=\sum_{j=1}^{n}\|u_j\|_{H^k(0,l_j)}^2\ k \in \mathbb{N}.$$
 \item[iv.] Consider the following characterization:
 $$H_r^{-1}(0,l_j)=\left(H_r^{1}(0,l_j)\right)'$$
 as the dual space of $H_r(0,l_j)$ with respect to the pivot space $L^2(0,l_j)$ and $\mathbb{H}_r^{-1}$ denotes the cartesian product of $H_r^{-1}(0,l_j)$.
\item[v.] For $k\in \mathbb{N}$, the derivative $\partial_x^ku$ means the row vector
\begin{align*}
\partial_x^k u:=(\partial_x^ku_1,...,\partial_x^ku_N).
\end{align*}
The derivative $\partial_t^ku$ is understood in the same way.
\item[vi.] Under the above notation, we have
\begin{align*}
 C^k\left([0,T];\mathbb{H}^s(\mathcal{T})\right)=\prod_{j=1}^NC^k\left([0,T];H^s(0,l_j)\right),\ k=0,1,2,... \text{ \ and \ }s \in \mathbb{R}
\end{align*}
as well as
\begin{align*}
 L^p\left(0,T;\mathbb{H}^s(\mathcal{T})\right)=\prod_{j=1}^NL^p\left(0,T;H^s(0,l_j)\right),\ p\geq 1 \text{ \ and \ }s \in \mathbb{R}.
\end{align*}
Hence, for $f \in C^k\left([0,T];\mathbb{H}^s(\mathcal{T})\right)$ and $q \in L^p\left(0,T;\mathbb{H}^s(\mathcal{T})\right)$ we can write
\begin{align*}
f=(f_1,...,f_N)\text{ \ and \ }q=(q_1,...,q_N)
\end{align*}
with
\begin{align*}
f_j \in C^k\left([0,T];H^s(0,l_j)\right)\text{ \ and \ }q_j \in L^p\left(0,T;H^s(0,l_j)\right),\ j=1,...N.
\end{align*}
\item[vii.] We will consider the spaces
$$\mathbb{H}^k_e(\mathcal{T})=\left\{u=(u_1,...,u_N)\in \mathbb{H}^k_r(\mathcal{T});\ u_j(0)=u_k(0),\ j,k=1,...,N\right\}\ k \in \mathbb{N},$$
$$\mathbb{H}^3_c(\mathcal{T})=\left\{u=(u_1,...,u_N)\in \mathbb{H}^3(\mathcal{T});\ u_j(0)=u_k(0),\ j,k=1,...,N\right\},$$
and
$$\mathbb{B}_{s,T}=C([0,T],\mathbb{H}^s(\mathcal{T}))\cap L^2(0,T,\mathbb{H}^{s+1}(\mathcal{T})),$$
with the norm
$$\displaystyle\|u\|_{\mathbb{B}_{s,T}}:=\|u\|_{C([0,T],\mathbb{H}^s(\mathcal{T}))}+\|u\|_{L^2(0,T,\mathbb{H}^{s+1}(\mathcal{T}))}=\max_{t\in [0,T]}\|u\|_{\mathbb{H}^s(\mathcal{T})}+\left(\int_0^T\|u(t,\cdot)\|_{\mathbb{H}^{s+1}}^2dt\right)^\frac{1}{2}.$$
When $s=0$ we write $\mathbb{B}_T$ instead $\mathbb{B}_{0,T}$.
\item [viii.] Once understand the appropriate spaces in each situation, we will use the notations $g=(g_1,...,g_N)$ and $p=(p_1,...,p_N)$.
\end{itemize}

\subsection{Main results and structure of the article} With the previous framework in hand, our goal here is to prove the local and global well-posedness for the initial boundary value problem (IBVP) \eqref{kdV}. The first main result establishes the local well-posedness for this system in sharp spaces.

\subsection{Main results and structure of the article} With the previous framework in hand, our goal here is to prove the local and global well-posedness for the initial boundary value problem (IBVP) \eqref{kdV}. The first main result establishes the local well-posedness for this system in sharp spaces.

\begin{lemma}\label{lemma for v}
 Let $(u^0,g_0, g,p)\in \mathbb{X}_{3,T}$, $f\in W^{1,1}(0,T;\mathbb{L}^2(\mathcal{T}))\cap L^2(0,T;\mathbb{H}^1(\mathcal{T}))$ and $v$ the solution of \eqref{v_j}. Then for $\beta=0,1,2,3$ we have:
 \begin{enumerate}
 \item[i.] The function
 \begin{align*}
 \begin{array}{rcl}
 [0,T]&\rightarrow&\mathcal{D}'(0,l_j)\\
 s&\mapsto&\partial_x^\beta v_j(s,\cdot)
 \end{array}
 \end{align*}
 is integrable in $[0,T]$.
 \item[ii.]For any $t \in [0,T]$ we have $\displaystyle\int_0^tv_j(s,\cdot)ds \in H^4(0,l_j)$ and for $\beta=0,1,2,3$ we have
 \begin{align*}
 \partial_x^\beta\int_0^tv_j(s,\cdot)ds=\int_0^t\partial_x^\beta v_j(s,\cdot)ds.
 \end{align*}
 \end{enumerate}
\end{lemma}
The previous lemma can give us the regularity for $u$.
\begin{proposition}\label{W.P._s=3}
Let $(u^0,g_0,g,p) \in \mathbb{H}^3(\mathcal{T})\times H^{\frac{3-1}{3}}(0,T)\times [H^\frac{3}{3}(0,T)]^N\times [H^{\frac{3+1}{3}}(0,T)]^N$ satisfying the linear compatibility conditions for $s=3$ and $f\in W^{1,1}(0,T;\mathbb{L}^2(\mathcal{T}))\cap L^2(0,T;\mathbb{H}^1(\mathcal{T}))$ be. For $j=1,...,N$ the function $t\mapsto u_j(t,\cdot)$ given by \eqref{u_j} satisfies the following items:
\begin{enumerate}
 \item[i.] $u_j \in C^1([0,T],L^2(0,l_j))\cap C([0,T];H^3(0,l_j))$;
 \item[ii.] $u\in \mathbb{B}_{3,T}=C([0,T];\mathbb{H}^3(\mathcal{T}))\cap L^2(0,T;\mathbb{H}^4(\mathcal{T}))$ is the unique classical solution to the problem \eqref{LkdV}; and
 \begin{equation*}
 \begin{split}
 \|u\|_{\mathbb{B}_{3,T}}\leq &C\left( \|u^0\|_{\mathbb{H}^3(\mathcal{T})}+\|g_0\|_{H^\frac{3-1}{3}(0,T)}+\|g\|_{[H^\frac{3}{3}(0,T)]^N}\right.\\
 &\left.+\|p\|_{[H^\frac{3+1}{3}(0,T)]^N}+\|f\|_{W^{1,1}(0,T;\mathbb{L}^2(\mathcal{T}))\cap L^2(0,T;\mathbb{H}^1(\mathcal{T}))}\right)
\end{split}
 \end{equation*}
 for some constant $C>0$.
 \item[iii.] $u \in H^1(0,T;\mathbb{H}^1(\mathcal{T}))$ and
 \begin{equation*}
 \begin{split}
 \|u\|_{H^1(0,T;\mathbb{H}^1(\mathcal{T}))}\leq& C\left(\|u^0\|_{\mathbb{H}^3(\mathcal{T})}+\|g_0\|_{H^\frac{3-1}{3}(0,T)}+\|g\|_{[H^\frac{3}{3}(0,T)]^N}\right.\\ &\left.+\|p\|_{[H^\frac{3+1}{3}(0,T)]^N}+\|f\|_{W^{1,1}(0,T;\mathbb{L}^2(\mathcal{T}))\cap L^2(0,T;\mathbb{H}^1(\mathcal{T}))}\right),
 \end{split}
 \end{equation*}
 for some constant $C>0$.
 \item [iv.] $\partial_x^ku_j\in L_x^\infty\left([0,l_j];H^\frac{3+1-k}{3}(0,T)\right)$ and
 \begin{align*}
 \|\partial_x^ku_j\|_{L_x^\infty\left([0,l_j];H^\frac{3+1-k}{3}(0,T)\right)}\leq& C\left(\|u^0\|_{\mathbb{H}^3(\mathcal{T})}+\|g_0\|_{H^\frac{3-1}{3}(0,T)}+\|g\|_{[H^\frac{3}{3}(0,T)]^N}\right.\\ &\left.+\|p\|_{[H^\frac{3+1}{3}(0,T)]^N}+\|f\|_{W^{1,1}(0,T;\mathbb{L}^2(\mathcal{T}))\cap L^2(0,T;\mathbb{H}^1(\mathcal{T}))}\right).
 \end{align*}
\end{enumerate}
\end{proposition}
\begin{proof}
Consider $j \in\{1,...,N\}$.
\vspace{0.2 cm}
\noindent\textbf{i.} Since $v_j\in C([0,T],L^2(0,l_j))$ it is immediate that by \eqref{u_j} that $u_j \in C^{1}([0,T],L^2(0,l_j))$.

\section{Local well-posedness of the nonlinear problem}\label{sec4}
Now, we are in a position to prove the first main result of the article, namely, the local well-posedness of the following problem
\begin{align}\label{NLKdV}
 \left\{
 \begin{array}{ll}
 \partial_t u_j+\partial_x [(a_j+1)u_j]+\partial_x^3 u_j+u_j\partial_x u_j=f_j,&t \in (0,T),\ x \in (0,l_j),\\
 u_j(t,0)=u_1(t,0),&t\in (0,T),\\
 \displaystyle\sum_{j=1}^N\partial_x^2 u_j(t,0)=-\alpha u_1(t,0)-\frac{N}{3}(u_1+a_1)^2(t,0)+g_0(t),&t \in (0,T)\\
 u_j(t,l_j)=p_j(t),\ \ \ \ \ \partial_x u_j(t,l_j)=g_j(t),&t\in (0,T),\\
 u_j(0,x)=u_j^0(x),&x \in (0,l_j),
 \end{array}
 \right.
\end{align}
where $j=1,...,N$ and $a$ is a given function. Remember that: for $s\geq 0$ and $T>0$ the space
\begin{align*}
\mathcal{X}_{s,T}=\left\{v \in \mathbb{B}_{s,T}^*;\ \partial_x^ku_j\in L_x^\infty(0,l_j;H^\frac{s+1-k}{3}(0,T)),\ k=0,1,2,\ j=1,2,...,N\right\}
\end{align*}
is a Banach space with the norm
\begin{align*}
\|v\|_{\mathcal{X}_{s,T}}=\|v\|_{\mathbb{B}_{s,T}^*}+\sum_{k=0}^2\|\partial_x^kv\|_{\prod_{j=1}^NL_x^\infty(0,l_j;H^\frac{s+1-k}{3}(0,T))}.
\end{align*}
Note that \eqref{kdV} is a particular case of \eqref{NLKdV} with $a=0$.
\begin{proof}[Proof of Theorem \ref{main}] Fix $\theta \in (0,T)$ and define $\Gamma_a:\mathcal{X}_{s,\theta}\rightarrow \mathcal{X}_{s,\theta}$ putting, for each $u \in \mathcal{X}_{s,\theta}$, $\Gamma_au$ as being the solution $v$ of
\begin{align*}
 \left\{
 \begin{array}{ll}
 \partial_t v_j+\partial_x [(a_j+1)v_j]+\partial_x^3 v_j=-u_j\partial_x u_j+f_j,&t \in (0,\theta),\ x \in (0,l_j),\ j=1,...,N,\\
 v_j(t,0)=v_1(t,0),&t\in (0,\theta),\ j=1,...,N,\\
 \displaystyle\sum_{j=1}^N\partial_x^2 v_j(t,0)=-\alpha v_1(t,0)-\frac{N}{3}(u_1+a_1)^2(t,0)+g_0(t),&t \in (0,\theta,)\\
 v_j(t,l_j)=p_j(t),\ \ \ \ \ \partial_x v_j(t,l_j)=g_j(t),&t\in (0,\theta),\ j=1,...,N,\\
 v_j(0,x)=u_j^0(x),&x \in (0,l_j),\ j=1,...,N.
 \end{array}
 \right.
\end{align*}
From Proposition \ref{W.P.LKdV-a} we have
\begin{align*}
& \|\Gamma_au\|_{\mathcal{X}_{s,\theta}}\leq C\left(\|u^0\|_{\mathbb{H}^s(\mathcal{T})}+\|g_0-\frac{N}{3}(u_1+a_1)^2(\cdot,0)\|_{H^\frac{s-1}{3}(0,\theta)}+\|g\|_{[H^\frac{s}{3}(0,\theta)]^N}\right.\\
&\left.+\|p\|_{[H^\frac{s+1}{3}(0,\theta)]^N}+\|u\partial_xu\|_{W^{\frac{s}{3},1}(0,\theta;\mathbb{L}^2(\mathcal{T}))\cap L^\frac{6}{6-s}(0,\theta;\mathbb{H}^\frac{s}{3}(\mathcal{T}))}+\|f\|_{W^{\frac{s}{3},1}(0,\theta;\mathbb{L}^2(\mathcal{T}))\cap L^\frac{6}{6-s}(0,\theta;\mathbb{H}^\frac{s}{3}(\mathcal{T}))} \right).
\end{align*}
Using Lemma \ref{lemma 2.4} we obtain
\begin{align*}
 \|\partial_xu v\|_{W^{\frac{s}{3},1}(0,\theta;\mathbb{L}^2(\mathcal{T}))\cap L^\frac{6}{6-s}(0,\theta;\mathbb{H}^\frac{s}{3}(\mathcal{T}))}\leq C(\theta^\frac{1}{2}+\theta^\frac{1}{3})\|u\|_{\mathbb{B}_{s,\theta}^*}\|u\|_{\mathbb{B}_{s,\theta}^*}.
\end{align*}
Furthermore, Lemma \ref{lemma 2.1} gives us
\begin{align*}
\left\|\frac{N}{3}(u_1+a_1)^2(\cdot,0)\right\|_{H^\frac{s-1}{3}(0,\theta)}&\leq C\theta^\alpha\frac{N}{3}\left\|(u_1+a_1)(\cdot,0)\right\|_{H^\frac{s+1}{3}(0,\theta)}^2\\
&\leq C\theta^\alpha\frac{N}{3}\|u+a\|_{\mathcal{X}_{s,\theta}}^2\\
&\leq \frac{4}{3}CN\theta^\alpha \left(\|u\|_{\mathcal{X}_{s,\theta}}^2+\|a\|_{\mathcal{X}_{s,\theta}}^2\right),
\end{align*}
where $\alpha>0$ is a constant. Thus,
\begin{align*}
 \|\Gamma_au\|_{\mathcal{X}_{s,\theta}}\leq& C\left(\|u^0\|_{\mathbb{H}^s(\mathcal{T})}+\|g_0\|_{H^\frac{s-1}{3}(0,T)}+\|g\|_{[H^\frac{s}{3}(0,T)]^N}\right.\\
&+\|p\|_{[H^\frac{s+1}{3}(0,T)]^N}+\|f\|_{W^{\frac{s}{3},1}(0,T;\mathbb{L}^2(\mathcal{T}))\cap L^\frac{6}{6-s}(0,T;\mathbb{H}^\frac{s}{3}(\mathcal{T}))}\\
&\left.+C(\theta^\frac{1}{2}+\theta^\frac{1}{3})\|u\|_{\mathcal{X}_{s,\theta}}^2+\frac{4}{3}CN\theta^\alpha \|u\|_{\mathcal{X}_{s,\theta}}^2+\frac{4}{3}CN\theta^\alpha\|a\|_{\mathcal{X}_{s,\theta}}^2 \right).
\end{align*}
Defining
\begin{align*}
r=&4C\left(\|u^0\|_{\mathbb{H}^s(\mathcal{T})}+\|g_0\|_{H^\frac{s-1}{3}(0,T)}+\|g\|_{[H^\frac{s}{3}(0,T)]^N}\right.\\
&\left.+\|p\|_{[H^\frac{s+1}{3}(0,T)]^N}+\|f\|_{W^{\frac{s}{3},1}(0,T;\mathbb{L}^2(\mathcal{T}))\cap L^\frac{6}{6-s}(0,T;\mathbb{H}^\frac{s}{3}(\mathcal{T}))}\right),
\end{align*}
it follows that, for $u \in B=\{v \in \mathcal{X}_{s,\theta};\ \|v\|_{\mathcal{X}_{s,\theta}}\leq r\}$,
\begin{align}\label{eq 3.2}
\|\Gamma_au\|_{\mathcal{X}_{s,\theta}}\leq \frac{r}{4}+C^2(\theta^\frac{1}{2}+\theta^\frac{1}{3})r^2+\frac{4}{3}C^2N\theta^\alpha r^2+\frac{4}{3}C^2N\theta^\alpha\|a\|_{\mathcal{X}_{s,T}}^2.
\end{align}

The proof of the previous theorem relies on several analytical tools combined with a fixed-point argument. In addition, a new notion of $s$-compatibility adapted to the graph structure is introduced (see Section \ref{sec2}), inspired by the single-edge KdV framework in \cite{BSZ 2003}. This theorem provides a sharp local well-posedness result, along with regularity properties, for the associated functions. Furthermore, we establish the existence of global solutions at the $H^s$-norm, which enables applications to control problems on star graphs and resolves several open questions that will be addressed in a forthcoming paper. The second main result of this work can be stated as follows.

\begin{theorem}\label{global_(0,3)}
Let $T>0$ and $\lambda\geq 1$ be. Consider $(u^0,g_0,g,p)\in \mathbb{X}_{s,T}^\lambda$, where $\mathbb{X}_{s,T}^\lambda$ is the set of $s$-compatible data
$$u^0 \in \mathbb{H}^{s}(\mathcal{T}), g_0 \in H^{\frac{s+\lambda-1}{3}}(0,T),\ g \in [H^\frac{s}{3}(0,T)]^N\quad \text{and}\quad p\in [H^\frac{s+\lambda+1}{3}(0,T)]^N.$$
Thus, the problem \eqref{kdV} has a unique solution $u\in \mathcal{X}_{s,T}$ which satisfies
\begin{align*}
	\|u\|_{\mathcal{X}_{s,T}}&\leq C\left(\|(u^0,g_0,g,p)\|_{0,\lambda}\right)\left(
	\begin{array}{c}
		\|u^0\|_{\mathbb{H}^s(\mathcal{T})}+\|g_0\|_{H^\frac{s+\lambda-1}{3}(0,T)}+\|g\|_{[H^\frac{s}{3}(0,T)]^N}+\|p\|_{[H^\frac{s+\lambda+1}{3}(0,T)]^N}
	\end{array}
	\right)
\end{align*}
where $C>0$ is a positive constant.
\end{theorem}
Together, these two theorems provide, for the first time, a comprehensive framework for the well-posedness of general graph structures with boundary controls. Moreover, they extend the classical result for a single KdV equation \cite{BSZ 2003} to star-shaped graphs composed of $N$ equations.
\vspace{0.2cm}

The paper is organized as follows. In Section \ref{sec2}, we introduce a general notion of $s$-compatibility for star graphs. Building on this framework, Section \ref{sec3} addresses the well-posedness of the linear system associated with \eqref{kdV}. After that, Section \ref{sec4} is devoted to proving the local well-posedness result for the system \eqref{kdV}, giving the proof of Theorem \ref{main}. The second main result, concerning global well-posedness, in the $H^s$-regime, for the system \eqref{kdV} is established in Section \ref{sec6}.

To prove Theorem~\ref{main}, the introduction states that it "relies on several analytical tools combined with a fixed-point argument." It also mentions that "a new notion of $s$-compatibility adapted to the graph structure is introduced" (Section~\ref{sec2}), "inspired by the single-edge KdV framework in \cite{BSZ 2003}." The paper’s organization further indicates the proof route: first introduce $s$-compatibility on star graphs (Section~\ref{sec2}); then treat "the well-posedness of the linear system associated with \eqref{kdV}" (Section~\ref{sec3}); and then Section~\ref{sec4} "is devoted to proving the local well-posedness result for the system \eqref{kdV}, giving the proof of Theorem~\ref{main}."

To prove the main theorem, the introduction states that it “relies on several analytical tools combined with a fixed-point argument.” It also mentions that “a new notion of $s$-compatibility adapted to the graph structure is introduced” (Section~\ref{sec2}), “inspired by the single-edge KdV framework in \cite{BSZ 2003}.” The paper’s organization further indicates the proof route: first introduce $s$-compatibility on star graphs (Section~\ref{sec2}); then treat “the well-posedness of the linear system associated with
\begin{align}\label{kdV}
			\begin{cases}
				\partial_t u_j+\partial_x u_j+\partial_x^3 u_j+u_j\partial_x u_j=0,&t \in (0,T),\ x \in (0,l_j),\ j=1,...,N\\
				u_j(t,0)=u_1(t,0),&t\in (0,T),\ j=1,...,N\\
				\displaystyle\sum_{j=1}^N\partial_x^2 u_j(t,0)=-\alpha u_1(t,0)-\frac{N}{3}u_1^2(t,0)+g_0(t),&t \in (0,T)\\
				u_j(t,l_j)=p_j(t),\ \ \ \ \ \partial_x u_j(t,l_j)=g_j(t),&t\in (0,T),\ j=1,...,N\\
				u_j(0,x)=u_j^0(x),&x \in (0,l_j),\ j=1,...,N
			\end{cases}
		\end{align}
” (Section~\ref{sec3}); and then Section~\ref{sec4} “is devoted to proving the local well-posedness result for the system above, giving the proof of the main theorem.”

\label{main} Let $T>0$, $l_j>0,\ j=1,...,N$, $s\in [0,3]$ be given. Suppose $f\in W^{\frac{s}{3},1}(0,T;\mathbb{L}^2(\mathcal{T}))\cap L^{\frac{6}{6-s}}(0,T;\mathbb{H}^\frac{s}{3}(\mathcal{T}))$ and
\begin{align*}
(u^0,g_0,g,p)\in \mathbb{H}^s(\mathcal{T})\times H^{\frac{s-1}{3}}(0,T)\times \left[H^\frac{s}{3}(0,T)\right]^{N}\times \left[H^{\frac{s+1}{3}}(0,T)\right]^N.
\end{align*}
satisfying the $s$-compatibility conditions, thus there exists $T^*\in (0,T)$ such that the IBVP
\begin{align}\label{kdV}
			\begin{cases}
				\partial_t u_j+\partial_x u_j+\partial_x^3 u_j+u_j\partial_x u_j=0,&t \in (0,T),\ x \in (0,l_j),\ j=1,...,N\\
				u_j(t,0)=u_1(t,0),&t\in (0,T),\ j=1,...,N\\
				\displaystyle\sum_{j=1}^N\partial_x^2 u_j(t,0)=-\alpha u_1(t,0)-\frac{N}{3}u_1^2(t,0)+g_0(t),&t \in (0,T)\\
				u_j(t,l_j)=p_j(t),\ \ \ \ \ \partial_x u_j(t,l_j)=g_j(t),&t\in (0,T),\ j=1,...,N\\
				u_j(0,x)=u_j^0(x),&x \in (0,l_j),\ j=1,...,N
			\end{cases}
		\end{align}
admits a unique solution $u \in \mathcal{X}_{s,T^*}$, where
$$\mathcal{X}_{s,T}=\left\{v \in \mathbb{B}^{*}_{s,T};\ \partial_x^ku_j\in L_x^\infty(0,l_j;H^\frac{s+1-k}{3}(0,T)),\ k=0,1,2,\ j=1,2,...,N\right\},$$
\begin{align*}
	\mathbb{B}_{s,T}^*:=\mathbb{B}_{s,T}\cap H^\frac{s}{3}(0,T;\mathbb{H}^1(\mathcal{T})).
\end{align*}

Question:
Let \(T>0\), \(l_j>0\) for \(j=1,\dots,N\), and \(s\in[0,3]\). Let \(\mathcal T\) be the star-shaped network with edges \((0,l_j)\), and write \(\mathbb L^2(\mathcal T)=\prod_{j=1}^N L^2(0,l_j)\) and \(\mathbb H^s(\mathcal T)=\prod_{j=1}^N H^s(0,l_j)\). Assume
\[
f\in W^{\frac{s}{3},1}(0,T;\mathbb L^2(\mathcal T))\cap L^{\frac{6}{6-s}}(0,T;\mathbb H^{\frac{s}{3}}(\mathcal T)),
\]
and
\[
(u^0,g_0,g,p)\in \mathbb H^s(\mathcal T)\times H^{\frac{s-1}{3}}(0,T)\times [H^{\frac{s}{3}}(0,T)]^N\times [H^{\frac{s+1}{3}}(0,T)]^N
\]
satisfy the \(s\)-compatibility conditions. Consider, for \(j=1,\dots,N\), the initial-boundary value problem
\[
\begin{cases}
\partial_t u_j+\partial_x u_j+\partial_x^3 u_j+u_j\partial_x u_j=0,& t\in(0,T),\ x\in(0,l_j),\\
u_j(t,0)=u_1(t,0),& t\in(0,T),\\
\displaystyle\sum_{j=1}^N \partial_x^2 u_j(t,0)=-\alpha u_1(t,0)-\frac{N}{3}u_1^2(t,0)+g_0(t),& t\in(0,T),\\
u_j(t,l_j)=p_j(t),\quad \partial_x u_j(t,l_j)=g_j(t),& t\in(0,T),\\
u_j(0,x)=u_j^0(x),& x\in(0,l_j).
\end{cases}
\]
Also set
\[
\mathbb B_{s,\tau}=C([0,\tau],\mathbb H^s(\mathcal T))\cap L^2(0,\tau;\mathbb H^{s+1}(\mathcal T)),
\]
\[
\mathbb B^*_{s,\tau}=\mathbb B_{s,\tau}\cap H^{\frac{s}{3}}(0,\tau;\mathbb H^1(\mathcal T)),
\]
and
\[
\mathcal X_{s,\tau}=\left\{v\in \mathbb B^*_{s,\tau}:\ \partial_x^k v_j\in L_x^{\infty}(0,l_j;H^{\frac{s+1-k}{3}}(0,\tau)),\ k=0,1,2,\ j=1,\dots,N\right\}.
\]
Which statement about existence and uniqueness is valid for this problem?

There exists \(T^*\in(0,T)\) such that the above initial-boundary value problem admits a unique solution \(u\in \mathcal X_{s,T^*}\).

Choices:
B. There exists \(T^*\in(0,T)\), depending only on \(T,s,N\) and the lengths \(l_j\), such that for every admissible \((u^0,g_0,g,p,f)\) satisfying the \(s\)-compatibility conditions, the above initial-boundary value problem admits a unique solution \(u\in \mathcal X_{s,T^*}\).
C. There exists \(T^*\in(0,T)\) such that the above initial-boundary value problem admits at most one solution in \(\mathcal X_{s,T^*}\).
D. For every \(T>0\), the above initial-boundary value problem admits a unique solution \(u\in \mathcal X_{s,T}\).
E. There exists \(T^*\in(0,T)\) such that the above initial-boundary value problem admits a unique solution \(u\in \mathbb B^*_{s,T^*}\).

The stem does not explicitly reveal the correct conclusion. It lists hypotheses and asks which well-posedness statement follows; the key features of the answer (local time interval, uniqueness, and the exact solution space) are not directly stated in the stem.

This is essentially a theorem-recall item: the stem reproduces the full hypotheses and the options are slight variants of the theorem's conclusion. It is not a pure verbatim restatement because the choices alter quantifiers, uniqueness, and function spaces, but it remains very close to the source theorem.

Some reasoning is needed to distinguish local vs global existence, uniqueness vs mere existence, and the precise solution space. However, the task mainly tests recognition of the exact theorem statement rather than substantial generative mathematical reasoning.

The distractors are plausible and mathematically meaningful: global-in-time instead of local, existence without uniqueness, incorrect dependence of the lifespan, and a weaker/incorrect solution space. These reflect common ways a theorem statement can be misstated.

A solid theorem-discrimination MCQ with strong distractors and little answer leakage, but it is still close to theorem restatement and only moderately tests genuine generative reasoning.


\begin{thm}\label{thm:1.1}
Given $\phi\in C(\partial\Omega)$, there exist a unique harmonic function $u_\phi$ on $\Omega$ and an extension $\Phi\in C(\clos{\Omega})$ of $\phi$ such that $\Phi-u_\phi\in H^1_0(\Omega)$.
\end{thm}

Throughout we suppose that $\Omega$ is a connected bounded open subset of $\mathbb{R}^d$, $d\ge 2$, with boundary $\partial\Omega$.
Given a function $\varphi\in C(\partial\Omega)$, the \textbf{\boldmath Dirichlet problem\unboldmath} asks to
\begin{equation}
 \tag*{$D(\varphi)$}
 \label{eq:D}
 \left\{
 \begin{aligned}
 &\text{find $u\in C(\overline{\Omega})$ which is harmonic on $\Omega$} \\
 &\text{such that $\restrict{u}{\partial\Omega}=\varphi$.}
 \end{aligned}
 \right.
\end{equation}
We call such a function $u$ a \textbf{\boldmath classical solution\unboldmath} of~\ref{eq:D}.
However, a classical solution does not always exist.

In this paper we consider a generalised solution of the Dirichlet problem which is described as follows.

\begin{equation}
  \tag*{$D(\varphi)$}
  \label{eq:D}
  \left\{
    \begin{aligned}
      &\text{find $u\in C(\clos{\Omega})$ which is harmonic on $\Omega$} \\
      &\text{such that $\restrict{u}{\partial\Omega}=\phi$.}
    \end{aligned}
  \right.
\end{equation}

Throughout we suppose that $\Omega$ is a connected bounded open subset of $\RR^d$, $d\ge 2$, with boundary $\partial\Omega$.
Given a function $\phi\in C(\partial\Omega)$, the \emphdef{Dirichlet problem} asks to
\begin{equation}
 \tag*{$D(\varphi)$}
 \label{eq:D}
 \left\{
 \begin{aligned}
 &\text{find $u\in C(\clos{\Omega})$ which is harmonic on $\Omega$} \\
 &\text{such that $\restrict{u}{\partial\Omega}=\phi$.}
 \end{aligned}
 \right.
\end{equation}
We call such a function $u$ a \emphdef{classical solution} of~\ref{eq:D}.
However, a classical solution does not always exist.

In this section we develop variational solutions of the Dirichlet problem on the basis of the following result~\cite[Corollary~4.3]{AtES24}.
\begin{thm}[Extension result]\label{thm:2.1}
Let $\phi\in C(\partial\Omega)$. Then there exist an extension $\Phi\in C(\clos{\Omega})$ of $\phi$ and $M \ge 0$ such that
\begin{equation}\label{eq:2.1}
 \abs*{\int_\Omega\Phi\Delta w\dx} \le M\norm{w}_{H^1(\Omega)}
\end{equation}
for all $w\in C^\infty_\cpt(\Omega)$. This means that $\Delta\Phi\in H^{-1}(\Omega)$, the dual space of $H^1_0(\Omega)$.
\end{thm}
Note that for $u\in H^1(\Omega)$ one has
\[
 \abs*{\int_\Omega u\Delta w\dx} = \abs*{-\int_\Omega\nabla u\nabla w\dx}\le\norm{u}_{H^1(\Omega)}\norm{w}_{H^1(\Omega)}
\]
for all $w\in C^\infty_\cpt(\Omega)$ and therefore $\Delta u\in H^{-1}(\Omega)$.

Let $\phi\in C(\partial\Omega)$ and $\Phi\in C(\clos{\Omega})$ be an extension of $\phi$ satisfying~\eqref{eq:2.1}.
By the Riesz--Fréchet representation theorem there exists a unique $v\in H^1_0(\Omega)$ such that
\[
 \int_\Omega \Phi\Delta w\dx = -\int_\Omega\nabla v\nabla w\dx = \int_\Omega v\Delta w\dx
\]
for all $w\in C^\infty_\cpt(\Omega)$. This means that $u:=\Phi-v$ satisfies $\Delta u=0$ weakly. Hence $u\in C^\infty(\Omega)$ by Weyl's theorem and $\Delta u=0$ in the classical sense;
that is, $u$ is a harmonic function.
We call $u$ the \emphdef{variational solution} of~\ref{eq:D} and set $u_\phi := u$.
To do so, we have to show that the function does not depend on the extension $\Phi$ of $\phi$. This follows from the next lemma.

Next we talk about the Dirichlet principle.
We start by mentioning that the variational solution $u_\phi$ is obtained by a minimising process.
In fact, let $\phi\in C(\partial\Omega)$ and $\Phi\in C(\clos{\Omega})$ be an extension of $\phi$ satisfying the estimate~\eqref{eq:2.1}, i.e.\ $\Delta\Phi\in H^{-1}(\Omega)$.
The solution $v\in H^1_0(\Omega)$ of $\Delta v=\Delta\Phi$ is the unique function $v\in H^1_0(\Omega)$ such that
\begin{equation}\label{eq:2.5}
 \frac{1}{2}\int_\Omega\abs{\nabla v}^2\dx + \langle\Delta\Phi,v\rangle = \min\Bigl\{\frac{1}{2}\int_\Omega\abs{\nabla w}^2\dx + \langle\Delta\Phi,w\rangle \setcolon w\in H^1_0(\Omega)\Bigr\}
\end{equation}
(see e.g.~\cite[Theorem~4.24]{AU23}).
Now we can characterise when $\int_\Omega\abs{\nabla u_\phi}^2\dx<\infty$; i.e.\ when $u_\phi$ has finite energy.
Since $u_\phi\in C^\infty(\Omega)\cap C_\bdd(\Omega)$, this is equivalent to saying that $u_\phi\in H^1(\Omega)$.

\begin{thm}\label{thm:2.7}
Let $\phi\in C(\partial\Omega)$. The following statements are equivalent.
\begin{romanenum}
\item\label{en:t2.7-i}
The variational solution $u_\phi$ of~\ref{eq:D} is an element of $H^1(\Omega)$.
\item\label{en:t2.7-ii}
There exists $\Phi\in C(\clos{\Omega})\cap H^1(\Omega)$ with $\restrict{\Phi}{\partial\Omega}=\phi$.
\end{romanenum}
In that case
\[
 \int_\Omega\abs{\nabla u_\phi}^2\dx = \min\Bigl\{\int_\Omega\abs{\nabla w}^2\dx \setcolon w\in H^1(\Omega),\ \Phi-w\in H^1_0(\Omega)\Bigr\}.
\]
\end{thm}
\begin{proof}
\begin{parenum}
\item[\ref{en:t2.7-ii}$\Rightarrow$\ref{en:t2.7-i}.]
Since $\Phi\in H^1(\Omega)$, $u_\phi=\Phi-v\in H^1(\Omega)$, where $v\in H^1_0(\Omega)$ is such that $\Delta v=\Delta\Phi$.
Moreover, by~\eqref{eq:2.5} and considering $W:=\{w\in H^1(\Omega)\setcolon \Phi-w\in H^1_0(\Omega)\}$,
\begin{align*}
 \frac{1}{2}[u_\phi,u_\phi] -\frac{1}{2}[\Phi,\Phi] &= \frac{1}{2}[v,v]-[\Phi,v] \\
 &= \min\Bigl\{\frac{1}{2}[\Phi-w,\Phi-w] - [\Phi,\Phi-w] \setcolon w\in W\Bigr\} \\
 &= \min\Bigl\{\frac{1}{2}[w,w] - \frac{1}{2}[\Phi,\Phi] \setcolon w\in W\Bigr\} \\
 &= \frac{1}{2} \min\Bigl\{[w,w] \setcolon w\in W\Bigr\}-\frac{1}{2}[\Phi,\Phi].
\end{align*}
Thus $[u_\phi,u_\phi] = \min\{[w,w]\setcolon w\in W\}$ and this property characterises $u_\phi$.
\item[\ref{en:t2.7-i}$\Rightarrow$\ref{en:t2.7-ii}.]
Let $\Phi\in C(\clos{\Omega})$ be such that $\restrict{\Phi}{\partial\Omega}=\phi$ and $\Delta\Phi\in H^{-1}(\Omega)$.
Let $v\in H^1_0(\Omega)$ be such that $\Delta v=\Delta\Phi$.
Then $u_\phi=\Phi-v$.
Since $u_\phi\in H^1(\Omega)$ by hypothesis, it follows that $\Phi\in H^1(\Omega)$.\qedhere
\end{parenum}
\end{proof}

If $\Omega$ is more regular, the following holds.
\begin{thm}\label{thm:2.9}
Assume that $\Omega$ has Lipschitz boundary and let $\phi\in C(\partial\Omega)$. The following statements are equivalent.
\begin{romanenum}
\item $u_\phi\in H^1(\Omega)$
\item $\phi\in H^{1/2}(\partial\Omega)$
\end{romanenum}
\end{thm}
Here, since $\Omega$ has Lipschitz boundary, $H^{1/2}(\partial\Omega)=\{\operatorname{tr}\Phi \setcolon \Phi\in H^1(\Omega)\}$, where $\operatorname{tr}\colon H^1(\Omega)\to L^2(\partial\Omega,\mathcal{H}^{d-1})$ is the trace operator and $\mathcal{H}^{d-1}$ denotes the $(d-1)$-dimensional Hausdorff measure.
If $\phi\in C(\partial\Omega)\cap H^{1/2}(\partial\Omega)$, there even exists a $\Phi\in C(\clos{\Omega})\cap H^1(\Omega)$ such that $\restrict{\Phi}{\partial\Omega}=\phi$.
We refer to~\cite[Theorem~1.2]{AtE19:dp} for this and a proof of Theorem~\ref{thm:2.9}.
A more general result on very rough domains is~\cite[Theorem~5.8]{AtES24}.

Theorem~\ref{thm:5.1} is contained in Landkof~\cite[p.\,243, lines 1--5]{La72}.
We will give a very different proof based on the following result. We assume that $d\ge 3$.
\begin{thm}\label{thm:5.2}
Let $z_0\in \partial\Omega$ be a singular point of $\Omega$.
Then there exists an open set $\Omega_0\subset\RR^d$ such that $\clos{\Omega_0}\subset\Omega\cup\{z_0\}$, $z_0\in\partial\Omega_0$ and $z_0$ is a singular point of $\Omega_0$.
\end{thm}
We postpone the proof of Theorem~\ref{thm:5.2} but note that in the case of Lebesgue's domain it readily follows from its construction.
More precisely, given the potential $V\colon D\to(0,\infty)$ from Lemma~\ref{lem:lebesgue-potential-basics}, we had $\Omega=\{x\in D\setcolon \frac{1}{2}<V(x)<2\}$. We can choose e.g.~$\Omega_0=\{x\in D\setcolon \frac{1}{c}<V(r,z)<c\}$ for any $c\in (1,2)$.
Now we can easily prove Theorem~\ref{thm:5.1} using the following criterion for a boundary point to be regular, see~\cite[Theorem~6.74]{AU23}.
\begin{prop}\label{prop:5.4}
Let $z_0\in \partial\Omega$. Then $z_0$ is regular if there exists a \emphdef{global $H^1$-barrier} $b$ at $z_0$; i.e.\ a function $b\in C(\clos{\Omega})\cap H^1(\Omega)$ such that $b(x)>0$ for all $x\in\clos{\Omega}\setminus\{z_0\}$, $b(z_0)=0$ and $-\Delta b\ge 0$ weakly on $\Omega$.
\end{prop}
\begin{proof}[Proof of Theorem~\ref{thm:5.1}]
Let $z_0\in \partial\Omega$ be a singular point for $\Omega$.
By Theorem~\ref{thm:5.2} there exists an open set $\Omega_0\subset\RR^d$ such that $\clos{\Omega_0}\subset\Omega\cup\{z_0\}$ and $z_0\in\partial\Omega_0$ is a singular point with respect to $\Omega_0$.
Let $0\le\phi\in C(\partial\Omega)$ be such that $\phi(z_0)=0$ and $\phi(w)>0$ at a regular point $w\in \partial\Omega$.
Then there exists $0\le\Phi_0\in C^\infty_\cpt(\RR^d)$ such that $\phi_0:=\restrict{\Phi_0}{\partial\Omega}$ satisfies $\phi_0\le\phi$ and $\phi_0(w)>0$.
Let $u_\phi$ and $u_{\phi_0}$ be the variational (equivalently Perron) solutions with respect to $\Omega$.
Then $u_\phi\ge u_{\phi_0}\ge 0$ on $\Omega$ by the maximum principle, and $u_{\phi_0}\ne 0$.
Moreover, $u_{\phi_0}\in H^1(\Omega)$ by Theorem~\ref{thm:2.7}.

We call $u_\varphi$ the \textbf{\boldmath variational solution\unboldmath} of~\ref{eq:D}.
It attains the given boundary value $\varphi$ in the Sobolev sense.
The existence of such a variational solution is difficult to prove and was an open problem for general $\varphi\in C(\partial\Omega)$ until recently.

In the first part of this paper we start with a simple proof of Theorem~\ref{thm:1.1}, making use of a nontrivial extension result that we shall take as given.
We then establish diverse properties of the variational solution defined as above.
We show that it is well-defined; i.e.\ $u_\varphi$ does not depend on the extension $\Phi$ of $\varphi$ and it coincides with the classical solution of~\ref{eq:D} whenever a classical solution exists.
Moreover, we show that the weak maximum principle is valid for variational solutions.
In Section~\ref{sec:3} we will see that $u_\varphi$ coincides with the Perron solution.
This notion was defined by Perron in 1923~\cite{Per1923} and has been a subject of intense research.
One subject is the way in which $u_\varphi(x)$ attains the value $\varphi(z)$ as $x\to z$.
So Theorem~\ref{thm:1.1} can be interpreted as a new description of the behaviour of the Perron solution at the boundary.

In the second part of the paper, from Section~\ref{sec:4} on, we investigate the behaviour of $u_\varphi(x)$ at a fixed singular point $z_0\in\partial\Omega$, i.e.\ a point where $\lim_{x\to z_0} u_\varphi(x)=\varphi(z_0)$ is violated for some $\varphi\in C(\partial\Omega)$.

In a single page paper Lebesgue~\cite{Leb1913} considered a potential in $\mathbb{R}^3$ of a thin rod modelled by a simple mass distribution concentrated along a line segment.
He chooses two suitable level sets of that potential and considers the bounded, open, simply connected region enclosed by these level sets.
We call this region \textbf{\boldmath Lebesgue's domain\unboldmath}.

Lebesgue's domain has a unique singular point $z_0$ which is the tip of an inward pointing cusp, the famous \textbf{\boldmath Lebesgue cusp\unboldmath}.
Taking the intersection of Lebesgue's domain with a small ball, one obtains a simply connected domain with connected boundary and a singular point.
This is of great interest since in $\mathbb{R}^2$ each simply connected domain is \textbf{\boldmath Dirichlet regular\unboldmath}; i.e.\ $u_\varphi$ is a classical solution for all $\varphi\in C(\partial\Omega)$.
Our point is to consider the original Lebesgue domain, whose boundary has two connected components.
It demonstrates two striking features of the generalised solution $u_\varphi$.

\begin{enumerate}[1.]
\item It is impossible to deduce from the regularity properties of $\varphi$ whether or not $u_\varphi$ is a classical solution.
The boundary of Lebesgue's domain is the disjoint union of two closed sets $\Gamma_2$ and $\Gamma_{\frac{1}{2}}$ (which are level sets of the considered potential).
If $\varphi$ is constant on each of these two closed sets with two different constants, $u_\varphi$ is not a classical solution.
To prove this, the variational definition of $u_\varphi$ will be most useful.

\item The irregular behaviour of $u_\varphi(x)$ as $x\to z_0$ is a non-local and unstable phenomenon.
For $\phi_0\equiv 0$ one has $u_{\phi_0}\equiv 0$, but if we perturb $\phi_0$ by a small compactly supported positive smooth bump in the neighbourhood of some point far away from $z_0$, then $u_\varphi$ is no longer a classical solution.
This can be seen easily at Lebesgue's domain, but we prove it for a singular point of an arbitrary domain in Section~\ref{sec:5}.
\end{enumerate}

As explained above, there seems to be no reasonable condition on the boundary function $\varphi$ to ensure that $u_\varphi$ is a classical solution of~\ref{eq:D} -- even though there are always many classical solutions.
The situation is different if we instead consider the question of when $u_\varphi$ has finite energy $\int_\Omega\abs{\nabla u_\varphi}^2$.
This is the case if and only if $\varphi$ has an extension $\Phi\in C(\overline{\Omega})\cap H^1(\Omega)$, and then $u_\varphi$ may be determined by the Dirichlet principle as the unique minimiser of the Dirichlet energy.
As an easy consequence, if $\varphi$ is Lipschitz continuous, then $u_\varphi$ has finite energy.
These results are proved in Section~\ref{sec:2} along with other properties of the variational solution.

In the first part of this paper we start with a simple proof of Theorem~\ref{thm:1.1}, making use of a \emph{nontrivial extension result} that we shall take as given.

\label{thm:1.1}
Given $\varphi\in C(\partial\Omega)$, there exist a unique harmonic function $u_\varphi$ on $\Omega$ and an extension $\Phi\in C(\overline{\Omega})$ of $\varphi$ such that $\Phi-u_\varphi\in H^1_0(\Omega)$.,

Question: Let \(\Omega\subset \mathbb{R}^d\) be a connected bounded open set with \(d\ge 2\) and boundary \(\partial\Omega\). Given a boundary function \(\varphi\in C(\partial\Omega)\), which existence-and-uniqueness statement holds? (Here \(H^1_0(\Omega)\) denotes the Sobolev space of \(H^1\)-functions with zero trace on \(\partial\Omega\).)

There exists exactly one harmonic function \(u_\varphi\) on \(\Omega\) such that there exists at least one extension \(\Phi\in C(\overline{\Omega})\) of \(\varphi\) with \(\Phi|_{\partial\Omega}=\varphi\) and \(\Phi-u_\varphi\in H^1_0(\Omega)\).

Choices:
B. There exists exactly one harmonic function \(u_\varphi\) on \(\Omega\) such that for every extension \(\Phi\in C(\overline{\Omega})\) of \(\varphi\) one has \(\Phi-u_\varphi\in H^1_0(\Omega)\).
C. There exists at least one harmonic function \(u\) on \(\Omega\) and at least one extension \(\Phi\in C(\overline{\Omega})\) of \(\varphi\) such that \(\Phi|_{\partial\Omega}=\varphi\) and \(\Phi-u\in H^1_0(\Omega)\).
D. There exists exactly one function \(u_\varphi\in C(\overline{\Omega})\) that is harmonic on \(\Omega\) and satisfies \(u_\varphi|_{\partial\Omega}=\varphi\); equivalently, one may choose \(\Phi=u_\varphi\), so that \(\Phi-u_\varphi\in H^1_0(\Omega)\).
E. For every extension \(\Phi\in C(\overline{\Omega})\) of \(\varphi\), there exists exactly one harmonic function \(u_\varphi\) on \(\Omega\) such that \(\Phi-u_\varphi\in H^1(\Omega)\).

The stem gives only the setup and definitions; it does not explicitly state the correct conclusion or provide a direct hint toward choice A over the nearby variants.

The item is close to theorem-recognition: the correct option is essentially a precise theorem statement under the given hypotheses. However, it is not a pure restatement because the alternatives vary quantifiers, uniqueness, and regularity in meaningful ways.

To choose correctly, the solver must distinguish subtle logical differences: existence vs uniqueness, existential vs universal choice of extension, and classical boundary regularity vs weaker harmonicity statements. That creates substantial reasoning pressure.

The distractors are mathematically plausible and target common failure modes: overclaiming boundary continuity (B), dropping uniqueness (C), mishandling quantifiers (D), and imposing unnecessary H^1/weak-solution structure (E). They are distinct and well-designed.

A strong MCQ with subtle, high-quality distractors and little answer leakage. Its main weakness is that it still functions partly as theorem-statement recognition rather than fully generative problem solving.


Question: Let \(\Omega\subset \mathbb{R}^d\) be a connected bounded open set with \(d\ge 2\) and boundary \(\partial\Omega\). Given a boundary function \(\varphi\in C(\partial\Omega)\), which existence-and-uniqueness statement holds? (Here \(H^1_0(\Omega)\) denotes the Sobolev space of \(H^1\)-functions with zero trace on \(\partial\Omega\).)

Choices:
B. There exists exactly one harmonic function \(u_\varphi\) on \(\Omega\) such that for every extension \(\Phi\in C(\overline{\Omega})\) of \(\varphi\) one has \(\Phi-u_\varphi\in H^1_0(\Omega)\).
C. There exists at least one harmonic function \(u\) on \(\Omega\) and at least one extension \(\Phi\in C(\overline{\Omega})\) of \(\varphi\) such that \(\Phi|_{\partial\Omega}=\varphi\) and \(\Phi-u\in H^1_0(\Omega)\).
D. There exists exactly one function \(u_\varphi\in C(\overline{\Omega})\) that is harmonic on \(\Omega\) and satisfies \(u_\varphi|_{\partial\Omega}=\varphi\); equivalently, one may choose \(\Phi=u_\varphi\), so that \(\Phi-u_\varphi\in H^1_0(\Omega)\).
E. For every extension \(\Phi\in C(\overline{\Omega})\) of \(\varphi\), there exists exactly one harmonic function \(u_\varphi\) on \(\Omega\) such that \(\Phi-u_\varphi\in H^1(\Omega)\).

The stem does not explicitly state the correct theorem or its key quantifier structure. It gives standard setup and notation, but the correct choice is not leaked; the test-taker must distinguish among subtly different existence/uniqueness formulations.

The item is close to a standard theorem statement about the weak Dirichlet problem, so it has some theorem-restatement flavor. However, it is not purely tautological because the options introduce meaningful variations in quantifiers, regularity, and uniqueness.

The question requires moderate reasoning: the solver must recognize why the exact weak formulation is correct and why stronger or weaker alternatives fail. Still, it mainly tests precise theorem recall/recognition rather than deeper generative inference.

The distractors are strong and mathematically plausible. They target common failure modes: over-strong universal quantification, loss of uniqueness, unjustified upgrade to classical boundary attainment, and confusion between H^1 and H^1_0.

A solid MCQ with no major answer leakage and high-quality distractors, but it remains somewhat close to theorem recognition rather than a strongly generative reasoning task.


\begin{lemma} \label{lem1}
For $k\geq 2$ we have that
$$f_k = (p_{k}-1) f_{k-1}.$$
\end{lemma}

\begin{theorem} \label{thm:main}
Let $p_1, \ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\prod_{i=1}^k p_i$ be their product.
For a given choice of residue classes, i.e., one residue class $r_i$ for each prime $p_i$, the number of available elements in $[1,P_k]$ is $a_k$ and the number of free elements is $f_k$.
In other words, for every choice of residue classes, there are $a_k$ integers $n$ with $1\leq n \leq P_k$ such that $\gamma(n)\leq 1$ and $f_k$ integers with $\gamma(n)=0$.
\end{theorem}

The sequence $A067549$ of The On-Line Encyclopedia of Integer Sequences \cite{oeis}, is defined as $(a_k)_{k \geq 1}$ with $a_k$ being the determinant of the $k \times k$ matrix whose diagonal contains the first $k$ prime numbers and all other elements are ones, i.e.,
$$ a_k = \det
 \begin{bmatrix}
2 & 1 & \ldots & 1 \\
1 & 3 & \ldots & 1 \\
\vdots & \vdots & \ddots & \vdots \\
1 & 1 & \ldots & p_k
\end{bmatrix}. $$
The first five elements, for $k=1, \ldots, 5$, of this sequence are
$$2, 5, 22, 140, 1448, \ldots,$$
see \cite{oeis} for a table of the first 300 elements of this sequence.
We relate this sequence to a concrete counting problem.
Let $p_1, \ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\prod_{i=1}^k p_i$ be their product.
We choose an arbitrary residue class $r_i$ for each prime $p_i$ and consider $k$ segments of arithmetic progressions restricted to the interval $[1,P_k]$, i.e., for $p_i$ and $r_i$ we consider
$$s_i = \{n : n= h\cdot p_i + r_i, h \in \NN_0 \} \cap [1,P_k]$$

For a given choice $r_1, \ldots, r_k$ of residue classes, let
$$\gamma(n)=\# \{p_i: n\equiv r_i \pmod{p_i} \}$$
for $1 \leq n \leq P_k$ be the number of residue classes in which $n$ is contained.
We introduce some notation:
\begin{itemize}
\item An integer $n$ is \emph{covered by a prime $p_i$} if it is contained in the residue class $r_i$.
\item An integer $n$ is \emph{free} if it is not contained in any residue class, i.e., $\gamma(n)=0$.
\item An integer $n$ is \emph{available} if it is contained in at most one residue class, i.e., $\gamma(n)\leq 1$.
\item An integer $n$ is \emph{occupied} if it is contained in at least two residue class, i.e., $\gamma(n)\geq 2$.
\end{itemize}
By definition, a free integer is also available.
Moreover, we define a second sequence $(f_k)_{k\geq 1}$ as follows:
$$ f_k = (-1)^{k} \det
 \begin{bmatrix}
 1 & 1 & \ldots & 1 & 1\\
2 & 1 & \ldots & 1& 1 \\
1 & 3 & \ldots & 1 & 1\\
\vdots & \vdots & \ddots & \vdots & \vdots\\
1 & 1 & \ldots & p_{k} & 1
\end{bmatrix}. $$
The first five elements of this sequence for $k=1, \ldots, 5$ are
$$1, 2, 8, 48, 480 \ldots,$$
We have the following main result:

\begin{abstract}
The sequence $A067549$ of The On-Line Encyclopedia of Integer Sequences \cite{oeis}, is defined as $(a_k)_{k \geq 1}$ with $a_k$ being the determinant of the $k \times k$ matrix whose diagonal contains the first $k$ prime numbers and all other elements are ones.
We relate this sequence to a concrete counting problem.
Choose an arbitrary residue class $r_i$ for each prime $p_i$ with $1 \leq i \leq k$ and set $P_k = \prod_{i=1}^k p_i$. We show that $a_k$ is the number of integers in $[1, P_k]$ that are contained in \emph{at most} one of the $k$ chosen residue classes.
Interestingly, we show that this sequence is closely related to the better known sequence $A005867$ for which we derive a novel characterisation in terms of determinants and which gives the number of integers in $[1, P_k]$ that are not contained in any of the $k$ residue classes.

\section{Introduction}
The sequence $A067549$ of The On-Line Encyclopedia of Integer Sequences \cite{oeis}, is defined as $(a_k)_{k \geq 1}$ with $a_k$ being the determinant of the $k \times k$ matrix whose diagonal contains the first $k$ prime numbers and all other elements are ones, i.e.,
$$ a_k = \det
 \begin{bmatrix}
2 & 1 & \ldots & 1 \\
1 & 3 & \ldots & 1 \\
\vdots & \vdots & \ddots & \vdots \\
1 & 1 & \ldots & p_k
\end{bmatrix}. $$
The first five elements, for $k=1, \ldots, 5$, of this sequence are
$$2, 5, 22, 140, 1448, \ldots,$$
see \cite{oeis} for a table of the first 300 elements of this sequence.
We relate this sequence to a concrete counting problem.
Let $p_1, \ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\prod_{i=1}^k p_i$ be their product.
We choose an arbitrary residue class $r_i$ for each prime $p_i$ and consider $k$ segments of arithmetic progressions restricted to the interval $[1,P_k]$, i.e., for $p_i$ and $r_i$ we consider
$$s_i = \{n : n= h\cdot p_i + r_i, h \in \NN_0 \} \cap [1,P_k]$$

For a given choice $r_1, \ldots, r_k$ of residue classes, let
$$\gamma(n)=\# \{p_i: n\equiv r_i \pmod{p_i} \}$$
for $1 \leq n \leq P_k$ be the number of residue classes in which $n$ is contained.
We introduce some notation:
\begin{itemize}
\item An integer $n$ is \emph{covered by a prime $p_i$} if it is contained in the residue class $r_i$.
\item An integer $n$ is \emph{free} if it is not contained in any residue class, i.e., $\gamma(n)=0$.
\item An integer $n$ is \emph{available} if it is contained in at most one residue class, i.e., $\gamma(n)\leq 1$.
\item An integer $n$ is \emph{occupied} if it is contained in at least two residue class, i.e., $\gamma(n)\geq 2$.
\end{itemize}
By definition, a free integer is also available.
Moreover, we define a second sequence $(f_k)_{k\geq 1}$ as follows:
$$ f_k = (-1)^{k} \det
 \begin{bmatrix}
 1 & 1 & \ldots & 1 & 1\\
2 & 1 & \ldots & 1& 1 \\
1 & 3 & \ldots & 1 & 1\\
\vdots & \vdots & \ddots & \vdots & \vdots\\
1 & 1 & \ldots & p_{k} & 1
\end{bmatrix}. $$
The first five elements of this sequence for $k=1, \ldots, 5$ are
$$1, 2, 8, 48, 480 \ldots,$$
We have the following main result:

Moreover, our method can be generalised to counting free and available integers in arbitrary sets of $k$ arithmetic progressions generated by $k$ different primes as shown in Section \ref{sec5}.

\begin{remark}
The problem of determining the number of free integers in the interval $[1,P_k]$ with residue classes $r_i=0$ for all $i$ is already mentioned by Dickson \cite[p 439]{dickson} in the first volume of his History of the Theory of Numbers. He refers to results by A. de Polignac, J. Deschamps and H.J.S Smith for which Martin \cite{martin} provides proofs in his manuscript.
Determining the number of available integers and the connection of the two counting problems to the determinants $a_k$ and $f_k$ are new to the best of our knowledge.
\end{remark}

We aim to understand the systematic intersections in the interval $[1,P_k]$ of $k$ arithmetic progressions generated by the first $k$ prime numbers $p_1, \ldots, p_k$ and arbitrary choice of residue classes $r_1, \ldots, r_k$ with $P_k = \prod_{i=1}^k p_i$.
In particular, we want to understand whether an integer $1\leq n \leq P_k$ is not covered by any residue class, is contained in at most one or in at least two residue classes.

Assume we have generated all $k$ arithmetic progression and intersected them with the interval $[1,P_k]$ obtaining the sets
$s_i$, $1\leq i \leq k$.
For each integer $1\leq n \leq P_k$ we can now ask in how many sets $s_i$ it is contained, i.e., compute $\gamma(n)$.
We denote the number of all available integers with $av(k)$, the number of free integers with $free(k)$ and the number of integers that are contained in two or more sets with $occ(k)$, i.e.
\begin{align*}
av(k) &:= \# \{n : \gamma(n) \leq 1 \} \\
free(k) &:=\# \{n : \gamma(n) = 0 \} \\
occ(k) &:=\# \{n : \gamma(n) > 1 \}
\end{align*}
To illustrate this, we consider the cases $k=1$ and $k=2$ with $p_1=2$, $p_2=3$ as well as $P_1=2$ $P_2=6$.
It is easy to see that $occ(1)=0$, $free(1)=1$ and $av(1)=2$, because the arithmetic progression for $p_1=2$ is the first progression we consider and it contains one element in $[1,2]$.

\begin{theorem} \label{thm2}
Let $p_{j_1}, \ldots, p_{j_k}$ be $k$ distinct prime numbers and let $P_k:=\prod_{i=1}^k p_{j_i}$ be their product.
For $k\geq 2$ and a given choice of residue classes, i.e., one residue class $r_i$ for each prime $p_i$, the number of available elements in $[1,P_k]$ is $A_k(p_{j_1}, \ldots, p_{j_k})$ and the number of free elements is $F_k(p_{j_1}, \ldots, p_{j_k})$.
In other words, for every choice of residue classes, there are $A_k(p_{j_1}, \ldots, p_{j_k})$ integers $n$ with $1\leq n \leq P_k$ such that $\gamma(n)\leq 1$ and $F_k(p_{j_1}, \ldots, p_{j_k})$ integers with $\gamma(n)=0$.
\end{theorem}

Moreover, our method can be generalised to counting free and available integers in arbitrary sets of $k$ arithmetic progressions generated by $k$ different primes as shown in Section \ref{sec5}.

\begin{remark}
The problem of determining the number of free integers in the interval $[1,P_k]$ with residue classes $r_i=0$ for all $i$ is already mentioned by Dickson \cite[p 439]{dickson} in the first volume of his History of the Theory of Numbers. He refers to results by A. de Polignac, J. Deschamps and H.J.S Smith for which Martin \cite{martin} provides proofs in his manuscript.
Determining the number of available integers and the connection of the two counting problems to the determinants $a_k$ and $f_k$ are new to the best of our knowledge.
\end{remark}

\begin{remark}
Lemma \ref{lem1} below gives a new characterisation of sequence $A005867$ from the The On-Line Encyclopedia of Integer Sequences \cite{oeis2} in terms of determinants.
\end{remark}

\begin{remark}
As a side product, the proof of Theorem \ref{thm:main} provides a quick way of calculating determinants of the form $a_k$ and $f_k$.
\end{remark}

The paper is organized as follows. In Section \ref{sec2} we derive structural results about $a_k$ and $f_k$. In Section \ref{sec3} we analyse the counting problem and obtain an equation that relates the available numbers after considering $k$ progressions to the available numbers after considering $k+1$ progressions. Theorem \ref{thm:main} is proven in Section \ref{sec4} and generalised in Section \ref{sec5}.

The post-theorem introduction does not give an explicit proof outline for Theorem~\ref{thm:main}, but it indicates the structure used to prove it: Section~\ref{sec2} "derive[s] structural results about $a_k$ and $f_k$"; Section~\ref{sec3} "analyse[s] the counting problem" and "obtain[s] an equation that relates the available numbers after considering $k$ progressions to the available numbers after considering $k+1$ progressions"; then "Theorem~\ref{thm:main} is proven in Section~\ref{sec4}" (and "generalised in Section~\ref{sec5}").

The post-theorem introduction does not give an explicit proof outline for the main theorem, but it indicates the structure used to prove it: next it derives structural results about $a_k$ and $f_k$; then it analyses the counting problem and obtains an equation that relates the available numbers after considering $k$ progressions to the available numbers after considering $k+1$ progressions; then it proves the main theorem (and later generalises it).

\label{thm:main}
Let $p_1, \ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\prod_{i=1}^k p_i$ be their product.
For a given choice of residue classes, i.e., one residue class $r_i$ for each prime $p_i$, the number of available elements in $[1,P_k]$ is $a_k$ and the number of free elements is $f_k$.
In other words, for every choice of residue classes, there are $a_k$ integers $n$ with $1\leq n \leq P_k$ such that $\gamma(n)\leq 1$ and $f_k$ integers with $\gamma(n)=0$.,

Question:
Fix a positive integer $k$. Let $p_1,\dots,p_k$ be the first $k$ prime numbers and let $P_k=\prod_{i=1}^k p_i$. For each $i$, choose one residue class $r_i \pmod{p_i}$, and for each integer $n$ with $1\le n\le P_k$ define
$$\gamma(n)=\#\{i\in\{1,\dots,k\}: n\equiv r_i \pmod{p_i}\}.$$
Call $n$ \emph{available} if $\gamma(n)\le 1$ and \emph{free} if $\gamma(n)=0$. Also define
$$a_k=\det\begin{bmatrix}
2 & 1 & \ldots & 1\\
1 & 3 & \ldots & 1\\
\vdots & \vdots & \ddots & \vdots\\
1 & 1 & \ldots & p_k
\end{bmatrix},$$
and
$$f_k=(-1)^k\det\begin{bmatrix}
1 & 1 & \ldots & 1 & 1\\
2 & 1 & \ldots & 1 & 1\\
1 & 3 & \ldots & 1 & 1\\
\vdots & \vdots & \ddots & \vdots & \vdots\\
1 & 1 & \ldots & p_k & 1
\end{bmatrix}.$$
Which statement holds for every choice of the residue classes $r_1,\dots,r_k$?

The number of available integers in $[1,P_k]$ is exactly $a_k$, and the number of free integers in $[1,P_k]$ is exactly $f_k$; equivalently,
$$\#\{n\in[1,P_k]:\gamma(n)\le 1\}=a_k\quad\text{and}\quad \#\{n\in[1,P_k]:\gamma(n)=0\}=f_k.$$

Choices:
B.
The number of available integers in $[1,P_k]$ is exactly $a_k$, and the number of free integers in $[1,P_k]$ is exactly $f_k$, provided the chosen residue classes $r_1,\dots,r_k$ are pairwise distinct as integers; equivalently,
$$\#\{n\in[1,P_k]:\gamma(n)\le 1\}=a_k\quad\text{and}\quad \#\{n\in[1,P_k]:\gamma(n)=0\}=f_k$$
for every such choice of pairwise distinct residues.
C.
The number of free integers in $[1,P_k]$ is exactly $f_k$ for every choice of the residue classes $r_1,\dots,r_k$; equivalently,
$$\#\{n\in[1,P_k]:\gamma(n)=0\}=f_k.$$
D.
For every choice of the residue classes $r_1,\dots,r_k$, the number of available integers in $[1,P_k]$ is at least $a_k$, and equality holds if and only if no integer of $[1,P_k]$ lies in more than two of the chosen residue classes; equivalently,
$$\#\{n\in[1,P_k]:\gamma(n)\le 1\}\ge a_k.$$
E.
The number of available integers in $[1,P_k]$ is exactly $f_k$, and the number of free integers in $[1,P_k]$ is exactly $a_k$; equivalently,
$$\#\{n\in[1,P_k]:\gamma(n)\le 1\}=f_k\quad\text{and}\quad \#\{n\in[1,P_k]:\gamma(n)=0\}=a_k.$$

dropped the available-count conclusion while keeping the free-count conclusion

exact recurrence-derived equality for available numbers

distinction between available and free counts

The stem introduces the objects and asks for a universal conclusion, but it does not state or strongly hint at the exact counting formula in choice A. The determinant definitions alone do not leak the answer.

The correct option is essentially the full theorem packaged as a choice, so the item is close to theorem recognition rather than an independently formulated problem. However, the presence of weaker, stronger, and tampered variants makes it more than a pure verbatim restatement.

There is some reasoning pressure because the student must distinguish the exact universal claim from a weaker true statement, an overstrong inequality/equality claim, and a swapped-count trap. Still, the item mainly tests recognition of the strongest correct statement rather than substantial generation or derivation.

The distractors are mathematically meaningful and target realistic failure modes: adding an unnecessary restriction (B), selecting a weaker true consequence (C), overstrengthening the conclusion (D), and confusing the two counts (E). They are distinct and plausible.

A solid theorem-recognition MCQ with strong distractors and no answer leakage, but only moderate generative depth because the correct choice is essentially the theorem itself.