\label{bc}

We note that
\begin{align}\label{bn}
\sum_{n=1}^\infty B(n)q^{n+1}&=q\sum_{n=1}^\infty\frac{q^{2n-1}}{(q;q^2)_n} \notag \\
&=\sum_{n=1}^\infty\frac{q^{2n}}{(q;q^2)_n},
\end{align}
here, an

\begin{theorem}\label{main theorem}
For $n>0$, we have
\begin{align}\label{main theorem eqn}
A(n)=B(n)=C(n+1)=\frac{1}{2}D(n+1),
\end{align}
where $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.
\end{theorem}

The grandfather of all partition identities is Euler's theorem \cite{andrews book}, namely,
\begin{align*}
A(n)=B(n),
\end{align*}
where $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts.

This theorem contains all the elements that would suggest generalizations, and, over the centuries, generalizations have been found in profusion. The Rogers-Ramanujan identities \cite[p.~104]{andrews book} and Schur's 1926 theorem \cite[p.~116]{andrews book} kicked off the twentieth century's contributions, consult Henry Alder's survey \cite{alder} for an account of some of the results in the late 20th century.

However, little has ever been added to Euler's theorem itself. In this paper, we shall add two further partition functions.

\subjclass[2020]{Primary 11P84, 05A17}
 \keywords{Euler's theorem, Partitions, Glaisher's bijection}
\maketitle
\pagenumbering{arabic}
\pagestyle{headings}
\begin{abstract}
Euler's theorem asserts that $A(n)=B(n)$ where $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts. In this paper, it is proved that for $n>0$,
\begin{align*}
A(n)=B(n)=C(n+1)=\frac{1}{2}D(n+1),
\end{align*}
where $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.

The grandfather of all partition identities is Euler's theorem \cite{andrews book}, namely,
\begin{align*}
A(n)=B(n),
\end{align*}
where $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts.

However, little has ever been added to Euler's theorem itself. In this paper, we shall add two further partition functions.

For example, if $n=6$, the four sets of partitions are following:
\begin{center}
\begin{tabular}{ p{2cm} p{3.5cm} p{3cm} p{4.1cm} p{4.1cm}}

On the other hand, we have
\begin{align}
\sum_{n=1}^\infty C(n)q^{n}&=\sum_{n=1}^\infty\frac{(-q;q)_nq^{2n}}{(q^{n+1};q)_n}\nonumber\\
&=\sum_{n=1}^\infty\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}} \label{c1n}\\
&=\sum_{n=1}^\infty\frac{q^{2n}}{(q;q^2)_n}.\label{cn}
\end{align}

Let us fix $C(0)=1$. From \eqref{c1n}, we have
\begin{align}\label{before p19}
\sum_{n=0}^\infty C(n)q^{n}&=\sum_{n=0}^\infty\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}}\nonumber\\
&=(q^2;q^2)_\infty\sum_{n=0}^\infty\frac{q^{2n}}{(q;q)_{2n}(q^{2n+2};q^2)_\infty}.
\end{align}
We now employ the following Euler's identity \cite[p.~19, (2.2.5)]{andrews book}
\begin{align}\label{euler identity}
\frac{1}{(t;q)_\infty}=\sum_{m=0}^\infty\frac{t^m}{(q,q)_m},
\end{align}
(with replacing $q$ by $q^2$ and then letting $t=q^{2n+2}$) in \eqref{before p19} so as to obtain
\begin{align}\label{series}
\sum_{n=0}^\infty C(n)q^{n}&=(q^2;q^2)_\infty\sum_{n=0}^\infty\frac{q^{2n}}{(q;q)_{2n}}\sum_{m=0}^\infty\frac{q^{2nm+2m}}{(q^2;q^2)_m}\nonumber\\
&=(q^2;q^2)_\infty\sum_{m,n=0}^\infty\frac{q^{2n+2nm+2m}}{(q;q)_{2n}(q^2;q^2)_m}\nonumber\\
&=(q^2;q^2)_\infty\sum_{m,n=0}^\infty\frac{1}{2}\left(1+(-1)^n\right)\frac{q^{n+nm+2m}}{(q;q)_{n}(q^2;q^2)_m}\nonumber\\
&=\frac{1}{2}(q^2;q^2)_\infty\sum_{m=0}^\infty\frac{q^{2m}}{(q^2;q^2)_m}\left\{\sum_{n=0}^\infty\frac{q^{n(m+1)}}{(q;q)_n}+\sum_{n=0}^\infty\frac{(-1)^nq^{n(m+1)}}{(q;q)_n}\right\}.
\end{align}
Upon invoking \eqref{euler identity} twice, once with letting $t=q^{m+1}$ and once with letting $t=-q^{m+1}$, and then substituting both resulting expressions in \eqref{series}, we conclude that
\begin{align}\label{3.4}
\sum_{n=0}^\infty C(n)q^{n}&=\frac{1}{2}(q^2;q^2)_\infty\sum_{m=0}^\infty\frac{q^{2m}}{(q^2;q^2)_m}\left\{\frac{1}{(q^{m+1};q)_\infty}+\frac{1}{(-q^{m+1};q)_\infty}\right\}\nonumber\\
&=\frac{1}{2}(-q;q)_\infty\sum_{m=0}^\infty\frac{q^{2m}}{(-q;q)_m}+\frac{1}{2}(q;q)_\infty\sum_{m=0}^\infty\frac{q^{2m}}{(q;q)_m}\nonumber\\
&=\frac{1}{2}\sum_{m=0}^\infty q^{2m}(-q^{m+1};q)_\infty+\frac{1}{2}(1-q),
\end{align}
where the last step follows upon again using \eqref{euler identity} with letting $t=q^2$.

\subsection{A Bijective proof of $A(n-1)=\frac{1}{2}D(n)$}
We can first remove the non-negative part condition for $D(n)$ as follows:
\begin{quote}
For $n\ge 2$, $D(n)$ counts the number of partitions of $n$ where only the smallest part can repeat at most twice and all other parts are distinct.
\end{quote}
In other words, a partition $\lambda=(\lambda_1,\ldots, \lambda_{\ell})$ counted by $D(n)$ satisfies
$$
\lambda_1>\lambda_2> \cdots > \lambda_{\ell-1}\ge \lambda_{\ell} \text{ if $\ell>1$}
$$
or
$\lambda=(\lambda_1)$ has only one part.

Let $M \le 2N$ be an even number, which can be written as
$$
M=2^k a \text{ for some odd integer $a$ and some $ k\ge 1$}.
$$
We apply Glaisher's bijection $\phi$ to $M$ and obtain $2^k$ parts of size $a$. Note that since
$
2^k a \le 2N,
$
$$
a \le N.
$$
Also, if $M\le N$, then $M$ can appear only once, so we get all distinct positive powers $2^k$ such that
\begin{equation}
2^k\le N/a. \label{power1}
\end{equation}
On the other hand, if $N< M\le 2N$, then
\begin{equation}
2^k> N/a, \label{power2}
\end{equation}
and $M$ can repeat.
Suppose $M$ appears $f$ many times. Upon applying Glaisher's bijection $\phi$ to $f$ copies of $M$, we obtain
$$
 f 2^{k} \text{ copies of $a$}.
$$
By writing $f$ as a binary expansion,
$$
f 2^k =(f_0 \,2^{0}+f_1 \, 2^{1} +\cdots ) 2^k,
$$
where $f_j$ is either $0$ or $1$ for $j\ge 0$. By \eqref{power2}, we see that each summand in the above expression represents a distinct power of $2$ greater than $N/a$, i.e.,
\begin{equation}
f_j \, 2^{j+k} >N/a. \label{power3}
\end{equation}
It follows from \eqref{power1} and \eqref{power3} that $a$ can appear with any multiplicity greater than $1$. This proves that the resulting partition is counted by $B(n)$.

For example, if $n=6$, the four sets of partitions are following:
\begin{center}
\begin{tabular}{ p{2cm} p{3.5cm} p{3cm} p{4.1cm} p{4.1cm}}

 \vspace{2mm}$A(6)$\newline  &  \vspace{2mm} $B(6)$ & \vspace{2mm} $C(7)$& \vspace{2mm} $D(7)$ \\

 $6$ \newline 5+1 \newline 4+2\newline 3+2+1   & 5+1 \newline 3+3 \newline 3+1+1+1 \newline 1+1+1+1+1+1  & 6+1\newline 4+3 \newline 4+2+1 \newline 2+2+2+1  & 0+0+7\newline 0+0+6+1\newline 0+0+5+2\newline 0+0+4+3\newline 0+0+4+2+1\newline 1+1+5\newline 1+1+2+3\newline 2+2+3
\end{tabular}
\end{center}

We conclude the introduction with the following remark.
\begin{remark}
In their recent paper \cite{ba}, M. El Bachraoui and the first author considered partitions with multiple appearances by the first part.  All parts were assumed to be positive. It would be a simple matter to extend the results of that paper to the case of non-negative parts in that this would add $(-q;q)_\infty$ to the generating functions in question.
\end{remark}

This paper is organised as follows. In section \ref{bc}, we provide the brief proof that $B(n)=C(n+1)$. In Section \ref{cd}, we prove that $C(n)=\frac{1}{2}D(n)$. We also provide bijective proofs our assertions in Theorem \ref{main theorem} in Section \ref{bijective proofs}.

The post-theorem introduction does not give a proof sketch beyond an outline of the paper: in Section~\ref{bc} they "provide the brief proof that $B(n)=C(n+1)$"; in Section~\ref{cd} they "prove that $C(n)=\frac{1}{2}D(n)$"; and in Section~\ref{bijective proofs} they "provide bijective proofs" of the assertions in Theorem~\ref{main theorem}.

The post-theorem introduction does not give a proof sketch beyond an outline of the paper: next they note that
\begin{align}\label{bn}
\sum_{n=1}^\infty B(n)q^{n+1}&=q\sum_{n=1}^\infty\frac{q^{2n-1}}{(q;q^2)_n} \notag \\
&=\sum_{n=1}^\infty\frac{q^{2n}}{(q;q^2)_n},
\end{align}
here, an
and also that
\begin{align}
\sum_{n=1}^\infty C(n)q^{n}&=\sum_{n=1}^\infty\frac{(-q;q)_nq^{2n}}{(q^{n+1};q)_n}\nonumber\\
&=\sum_{n=1}^\infty\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}} \label{c1n}\\
&=\sum_{n=1}^\infty\frac{q^{2n}}{(q;q^2)_n}.\label{cn}
\end{align}

Comparing \eqref{bn} and \eqref{cn}, we see that $B(n)=C(n+1)$.

\section{Proof that $C(n)=\frac{1}{2}D(n)$}\label{cd}

Let us fix $C(0)=1$. From \eqref{c1n}, we have
\begin{align}\label{before p19}
\sum_{n=0}^\infty C(n)q^{n}&=\sum_{n=0}^\infty\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}}\nonumber\\
&=(q^2;q^2)_\infty\sum_{n=0}^\infty\frac{q^{2n}}{(q;q)_{2n}(q^{2n+2};q^2)_\infty}.
\end{align}
They then prove that $C(n)=\frac{1}{2}D(n)$, and finally they provide bijective proofs as follows:
\label{bijective proofs}

\subsection{A Bijective proof of $A(n-1)=\frac{1}{2}D(n)$}
We can first remove the non-negative part condition for $D(n)$ as follows:
\begin{quote}
For $n\ge 2$, $D(n)$ count

\label{main theorem}
For $n>0$, we have
\begin{align}\label{main theorem eqn}
A(n)=B(n)=C(n+1)=\frac{1}{2}D(n+1),
\end{align}
where $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.,

Question: For each integer \(n>0\), let \(A(n)\) be the number of partitions of \(n\) into distinct parts, and let \(B(n)\) be the number of partitions of \(n\) into odd parts. Also, for an integer \(m\), let \(C(m)\) be the number of partitions of \(m\) whose largest part is even and whose parts not exceeding half of the largest part are distinct, and let \(D(m)\) be the number of partitions of \(m\) into non-negative parts such that the smallest part appears exactly twice and no other part is repeated. Which statement is equivalent to the equality \(A(n)=B(n)\) for all \(n>0\)?

For every \(n>0\), \(A(n)=B(n)=C(n+1)=\tfrac12 D(n+1)\).

Choices:
B. For every \(n>0\), \(A(n)=B(n)=C(n)=\tfrac12 D(n)\).
C. For every \(n>0\), \(A(n)=B(n)=C(n+1)\).
D. For every \(n>0\), \(A(n)=B(n)=C(n+1)=D(n+1)\).
E. For every \(n>0\), \(A(n)=B(n)=C(n+1)=\tfrac12 D(n)\).

The stem defines the partition functions and asks for an equivalent statement, but it does not explicitly reveal the needed index shift or the factor of 1/2. The correct answer is not stated or strongly hinted at in the wording.

The item is close to asking for the exact theorem-level equivalence rather than a genuinely new conclusion. It is not a pure restatement of A(n)=B(n), since the C and D relations must be matched correctly, but it still mainly tests recognition of the precise formulation.

Some reasoning is needed to distinguish subtle variants involving shifted indices, a missing factor, and a weaker true statement. However, the task is still largely theorem recognition/verification rather than substantial generative mathematical reasoning.

The distractors are plausible and well targeted: off-by-one shifts, omission of the 1/2 factor, and inclusion of a weaker true statement. These reflect realistic mathematical errors and are meaningfully distinct.

A solid MCQ with no real answer leakage and strong distractors, but it leans toward recalling the exact theorem statement rather than forcing deep generative reasoning.


\label{t.main}
	Let $n,s,\ell,c$ be positive integers such that $n=2s+c=3s-\ell$, and $c,\ell\in [s-1]$. Then $$e(n, s) = \max\big\{|\mathcal{P}(s, \ell)|, |\mathcal{P}'(\ell)|, |\mathcal{Q}(s, \ell)|, |\mathcal{W}(s, \ell)|\big\}.$$ Moreover, if $s \geq 3$, ${\mathcal F}\subset 2^{[n]}$ is shifted, has no $s$-matching and $|{\mathcal F}|=e(n,s)$, then ${\mathcal F}$ must coincide with one of the families above.

\begin{thm}[\cite{FK9,FK8}]\label{thmfk} \ $e(sm+s-\ell,s) = |\pp(s,m,\ell)|$ holds for
\begin{align*} &\mathrm{(i)}\ \ \ \ \ell = 2, \\
    &\mathrm{(ii)}\ \ \  m=1,\\
     &\mathrm{(iii)}  \ \  s\ge \ell m+3\ell+3.
\end{align*}
\end{thm}

\begin{gypo}[\cite{FK9}]\label{conj1}
Suppose that $s\ge 2, m\ge 1$, and $n = sm+s-\ell$ for some integer $0<\ell\le \lceil \frac s2\rceil$. Then
\begin{equation}\label{eq007} e(sm+s-\ell,s) = |\pp(s,m,\ell)|.
\end{equation}
\end{gypo}

Let $[n] := \{1,2,\ldots, n\}$ and, more generally, $[a,b]=\{a,a+1,\ldots, b\}$. For a set $X$ and an integer $k$, let  $2^{X}$, ${X\choose k}$ and ${X\choose \geq k}$ stand for the power set of $X$, the set of its $k$-element subsets and the set of its subsets with size at least $k$, respectively. Any collection of sets is called a {\it family.} A {\it matching} is a collection of pairwise disjoint sets. An {\it $s$-matching} is a matching of size $s$. Given a family ${\mathcal F},$ its {\it matching number}
$\nu(\mathcal F)$ is the size of the largest matching in ${\mathcal F}$.

One of the classical topics in extremal set theory is the study of {\it intersecting} families, that is, families with matching number $1$. Erd\H os, Ko and Rado~\cite{EKR} showed that the largest intersecting family ${\mathcal F}\subset 2^{[n]}$ has size at most $2^{n-1}$, and that for $n\ge 2k$ the largest intersecting family ${\mathcal F}\subset {[n]\choose k}$ has size ${n-1\choose k-1}.$ In the several years that followed, Erd\H os asked for the size of the largest family avoiding an $s$-matching. Let us introduce the following two quantities.
\begin{align*}
    e(n,s)&=\max\big\{|{\mathcal F}|: {\mathcal F}\subset 2^{[n]}, \nu({\mathcal F})<s\big\},\\
    e_k(n,s)&=\max\Big\{|{\mathcal F}|: {\mathcal F}\subset {[n]\choose k}, \nu({\mathcal F})<s\Big\}.
\end{align*}
\subsection{The uniform case}
The value of $e_k(n,s)$ is the subject of the famous Erd\H os Matching Conjecture. Let us define the families $\aaa_i^{(k)}(n,s):$
\begin{equation}\label{eq0011} \aaa_i^{(k)}(n,s-1) := \Bigl\{F\in {[n]\choose k}:|F\cap [si-1]|\ge i\Bigr\}, \ \ \ 1\le i\le k.\end{equation}
It is not difficult to see that $\nu(\aaa_i^{(k)}(n,s-1)\le s-1$ for any $n,s$.
\begin{gypo}[Erd\H os Matching Conjecture \cite{E}]\label{conj2} For $n\ge sk$
\begin{equation}\label{eq008} e_k(n,s) = \max\bigl\{|\aaa_1^{(k)}(n,s-1)|,|\aaa_k^{(k)}(n,s-1)|\bigr\}.
\end{equation}
\end{gypo}
Let us comment on the statement of the conjecture. First, the condition $n\ge sk$ is required in order for ${[n]\choose k}$ to contain an $s$-matching. Second, note that the conjecture suggests that only two out of $k+1$ families $\aaa_i$ could be extremal. This is different from some other similar-looking questions, notably, the Complete $t$-Intersection Theorem for ${[n]\choose k}$ \cite{AK}. There, all families could be extremal depending on the parameters $n,k,t.$

The conjecture (\ref{eq008}) is known to be true for $k= 2$ \cite{EG}, and $k=3$ \cite{LM,F11}. In fact, an elegant proof of Frankl of the $k=2$ case of the conjecture, initially proven by Erd\H os and Gallai, will play an important role in this paper. We will recite it later.

The conjecture was proved for $n\ge n_0(s,k)$ by Erd\H os \cite{E}. Many researchers worked on extending the range in which the conjecture is valid:  \cite{E}, \cite{BDE}, \cite{HLS}, \cite{FLM},  \cite{F4}, \cite{FK21}. Notably, in the last two papers, it is proved that
\begin{equation}\label{eq009} e_k(n,s) = |\aaa_1|={n\choose k}-{n-s+1\choose k} \end{equation}
for $n\ge (2s+1)k-s$ and for ($n\ge \frac 53sk$, $s\ge s_0$), respectively. In \cite{F7} and \cite{KoKu}, the authors studied the problem from the other end and showed that $\aaa_k$ is extremal for ($n\le s(k+1/100k)$, $s\ge ck^3$ with some absolute $c>0$). We refer the reader to \cite{aletal, FK21} for the connections of the Erd\H os Matching Conjecture and other questions, such as Dirac thresholds and small deviations in probability theory. In \cite{HLS}, \cite{K49}, the multi-family variant of the EMC was addressed.  In \cite{FK6}, a Hilton--Milner type stability result for the EMC is obtained.

\subsection{The non-uniform case}
The study of $e(n,s)$ was also initiated by Erd\H os at around the same time. The behavior of $e(n,s)$ heavily depends on $n\ ({\rm mod\ } s)$.  Answering a question of Erd\H os, Kleitman proved the following theorem.
\begin{thm}[Kleitman \cite{Kl}]\label{thmkl}
\begin{align}\label{eq001} e(sm-1,s) &= \sum_{t=m}^{sm-1}{sm-1\choose t},\\
\label{eq002} e(sm,s) &= {sm-1\choose m}+\sum_{t=m+1}^{sm}{sm\choose t}.
\end{align}
\end{thm}
The matching example for the first case is the family ${[n]\choose \ge m}$ of all subsets of $[n]$ of size at least $m$. It is also  not difficult to see that $e(sm,s) = 2e(sm-1,s)$. In general, $e(n+1,s)\ge 2e(n,s)$ because of the {\it doubling} construction. Given a family ${\mathcal F}\subset 2^{[n]}$ with $\nu({\mathcal F})<s$, we may construct the {\it doubling} ${\mathcal F}'$ of ${\mathcal F}$ as follows: ${\mathcal F}' = \{F\subset [n+1]: F\cap [n]\in {\mathcal F}\}$. It is easy to see that $\nu({\mathcal F}')<s$ as well.

In \cite{FK9}, Frankl and the first author put forth the following conjecture.
\begin{opr}\label{def3} Let $n = sm+s-\ell$, $0<\ell\le s$. Set
\[{\mathcal P}(s,m,\ell) := \bigl\{P\subset 2^{[n]}: |P|+|P\cap [\ell-1]|\ge m+1\bigr\}.\]
\end{opr}
It is not difficult to check (cf. \cite{FK9}) that $\nu( \mathcal P(s,m,\ell))<s.$
\begin{gypo}[\cite{FK9}]\label{conj1}
Suppose that $s\ge 2, m\ge 1$, and $n = sm+s-\ell$ for some integer $0<\ell\le \lceil \frac s2\rceil$. Then
\begin{equation}\label{eq007} e(sm+s-\ell,s) = |{\mathcal P}(s,m,\ell)|.
\end{equation}
\end{gypo}
They confirmed the conjecture for a variety of cases:
\begin{thm}[\cite{FK9,FK8}]\label{thmfk} \ $e(sm+s-\ell,s) = |{\mathcal P}(s,m,\ell)|$ holds for
\begin{align*} &\mathrm{(i)}\ \ \ \ \ell = 2, \\
    &\mathrm{(ii)}\ \ \  m=1,\\
     &\mathrm{(iii)}  \ \  s\ge \ell m+3\ell+3.
\end{align*}
\end{thm}
The case of $\ell=2,s=3$ was solved much earlier by Quinn \cite{Q} in his PhD dissertation. We also note that a sum-type inequality for $s$ families avoiding a matching is much easier to obtain, see \cite{FK6}.

Let us return to the definition of $\mathcal P(s,m,\ell)$. The two main uniformities to understand the construction are $m$ and $m+1$. One sees that the family contains all sets of size $m+1$, as well as all sets of size $m$ that intersect $[\ell-1]$, i.e., the family $\aaa_1^{(m)}(n,\ell-1)$. Thus, in fact, the sets of uniformity $m$ form a family with no matching of size $\ell$. We should note here that the sets in $\mathcal P(s,m,\ell)\cap {[n]\choose \le m}$, a bulk of which is  $\aaa_1^{(m)}(n,\ell-1)$, could be replaced by any other family in ${[n]\choose m}$ with no matching of size $\ell$. As a result, we would obtain a construction of a family in $2^{[n]}$ with no $s$-matching.

Here, one sees that the problems of determining $e(n,s)$ and $e_k(n,s)$ are, in fact, closely connected. In particular, let us give  the reason for the constraint $\ell\le s/2$ in Conjecture~\ref{conj1}: in this regime, $n\ge 2\ell m,$ and the result of Frankl \cite{F4} guarantees that the largest family in ${[n]\choose m}$ with no matching of size $\ell$ is indeed $\aaa_1^{(m)}(n,\ell-1)$.

Incidentally, one could also interpret the extremal family for $n=sm = sm+s-s$ in the same way: the extremal example consists of all sets of size at least $m+1$, as well as of the family of sets of size $m$ with no $s$-matching. The construction for the sets of uniformity $m$ is, however, different: it is $\aaa_k^{(m)}(n,s-1)$. Actually, the situation is even more complicated, as is shown in \cite{FK9}: already for $n = sm+1$ there is a construction in which some  $(m+1)$-element sets are missing and that is bigger than constructions with all $(m+1)$-element sets present. Namely, it is the doubling construction.

Concluding the discussion, the authors of \cite{FK9} state that it is hard to formulate a general conjecture, and that they had already struggled with the case $n=2s+t$ for some small values of $t$.
\subsection{Main result}
In this paper, we find the value of $e(n,s)$ for all $n,s$, provided $n\le 3s.$ The easy case $n<2s$ is solved in Theorem~\ref{thmfk}. The cases $n=2s$ and $n=3s$ are answered in \eqref{eq002}. Thus, we are left to deal with the cases $2s<n<3s$. For the rest of the paper, unless otherwise stated, we use notation $n=2s+c=3s-\ell$, where $c, \ell\in[s-1]$, $c+\ell=s$.

In order to state the result, we describe the prospective extremal families. We have already seen the first example.
$$\mathcal{P}(s, \ell) := \mathcal P(s,2,\ell)=\big\{P \in 2^{[n]} : |P| + |P \cap [\ell - 1]| \geq 3\big\}.$$
The second family also contains all sets of size at least $3$, but differs on the $2$-uniform layer. It has the other extremal example for $2$-uniform families avoiding an $\ell$-matching: a clique on $2\ell-1$ vertices.
$$\mathcal{P}'(s, \ell) := {[n] \choose \geq 3} \cup {[2\ell-1] \choose 2}.$$
The third family misses some $3$-element sets, and instead has a larger $2$-uniform layer. It is also a clique. $$\mathcal{Q}(s, \ell) := {[n] \choose \geq 3} \cup {[s+\ell-1] \choose 2} \setminus {[s+\ell, n] \choose 3}.$$
Finally, the fourth family is an iterated doubling of the extremal family for $n=2s-1$.
$$\mathcal{W}(s, \ell) := \big\{P \in 2^{[n]} : |P \cap [2s - 1]| \geq 2\big\}.$$ We note that none of these families contain $0$- or $1$-element sets. In the next section, we show that all these families indeed avoid $s$-matchings.

The main result of the paper is the following theorem.

One of the classical topics in extremal set theory is the study of {\it intersecting} families, that is, families with matching number $1$. Erd\H os, Ko and Rado~\cite{EKR} showed that the largest intersecting family ${\mathcal F}\subset 2^{[n]}$ has size at most $2^{n-1}$, and that for $n\ge 2k$ the largest intersecting family ${\mathcal F}\subset {[n]\choose k}$ has size ${n-1\choose k-1}.$ In the several years that followed, Erd\H os asked for the size of the largest family avoiding an $s$-matching. Let us introduce the following two quantities.
\begin{align*}
 e(n,s)&=\max\big\{|{\mathcal F}|: {\mathcal F}\subset 2^{[n]}, \nu({\mathcal F})<s\big\},\\
 e_k(n,s)&=\max\Big\{|{\mathcal F}|: {\mathcal F}\subset {[n]\choose k}, \nu({\mathcal F})<s\Big\}.
\end{align*}
\subsection{The uniform case}
The value of $e_k(n,s)$ is the subject of the famous Erd\H os Matching Conjecture. Let us define the families $\aaa_i^{(k)}(n,s):$
\begin{equation}\label{eq0011} \aaa_i^{(k)}(n,s-1) := \Bigl\{F\in {[n]\choose k}:|F\cap [si-1]|\ge i\Bigr\}, \ \ \ 1\le i\le k.\end{equation}
It is not difficult to see that $\nu(\aaa_i^{(k)}(n,s-1)\le s-1$ for any $n,s$.
\begin{gypo}[Erd\H os Matching Conjecture \cite{E}]\label{conj2} For $n\ge sk$
\begin{equation}\label{eq008} e_k(n,s) = \max\bigl\{|\aaa_1^{(k)}(n,s-1)|,|\aaa_k^{(k)}(n,s-1)|\bigr\}.
\end{equation}
\end{gypo}
Let us comment on the statement of the conjecture. First, the condition $n\ge sk$ is required in order for ${[n]\choose k}$ to contain an $s$-matching. Second, note that the conjecture suggests that only two out of $k+1$ families $\aaa_i$ could be extremal. This is different from some other similar-looking questions, notably, the Complete $t$-Intersection Theorem for ${[n]\choose k}$ \cite{AK}. There, all families could be extremal depending on the parameters $n,k,t.$

\subsection{The non-uniform case}
The study of $e(n,s)$ was also initiated by Erd\H os at around the same time. The behavior of $e(n,s)$ heavily depends on $n\ ({\rm mod\ } s)$. Answering a question of Erd\H os, Kleitman proved the following theorem.
\begin{thm}[Kleitman \cite{Kl}]\label{thmkl}
\begin{align}\label{eq001} e(sm-1,s) &= \sum_{t=m}^{sm-1}{sm-1\choose t},\\
\label{eq002} e(sm,s) &= {sm-1\choose m}+\sum_{t=m+1}^{sm}{sm\choose t}.
\end{align}
\end{thm}
The matching example for the first case is the family ${[n]\choose \ge m}$ of all subsets of $[n]$ of size at least $m$. It is also not difficult to see that $e(sm,s) = 2e(sm-1,s)$. In general, $e(n+1,s)\ge 2e(n,s)$ because of the {\it doubling} construction. Given a family ${\mathcal F}\subset 2^{[n]}$ with $\nu({\mathcal F})<s$, we may construct the {\it doubling} ${\mathcal F}'$ of ${\mathcal F}$ as follows: ${\mathcal F}' = \{F\subset [n+1]: F\cap [n]\in {\mathcal F}\}$. It is easy to see that $\nu({\mathcal F}')<s$ as well.

In \cite{FK9}, Frankl and the first author put forth the following conjecture.
\begin{opr}\label{def3} Let $n = sm+s-\ell$, $0<\ell\le s$. Set
\[{\mathcal P}(s,m,\ell) := \bigl\{P\subset 2^{[n]}: |P|+|P\cap [\ell-1]|\ge m+1\bigr\}.\]
\end{opr}
It is not difficult to check (cf. \cite{FK9}) that $\nu( \mathcal P(s,m,\ell))<s.$
\begin{gypo}[\cite{FK9}]\label{conj1}
Suppose that $s\ge 2, m\ge 1$, and $n = sm+s-\ell$ for some integer $0<\ell\le \lceil \frac s2\rceil$. Then
\begin{equation}\label{eq007} e(sm+s-\ell,s) = |{\mathcal P}(s,m,\ell)|.
\end{equation}
\end{gypo}
They confirmed the conjecture for a variety of cases:
\begin{thm}[\cite{FK9,FK8}]\label{thmfk} \ $e(sm+s-\ell,s) = |{\mathcal P}(s,m,\ell)|$ holds for
\begin{align*} &\mathrm{(i)}\ \ \ \ \ell = 2, \\
 &\mathrm{(ii)}\ \ \ m=1,\\
 &\mathrm{(iii)} \ \ s\ge \ell m+3\ell+3.
\end{align*}
\end{thm}
The case of $\ell=2,s=3$ was solved much earlier by Quinn \cite{Q} in his PhD dissertation. We also note that a sum-type inequality for $s$ families avoiding a matching is much easier to obtain, see \cite{FK6}.

Let us return to the definition of $\mathcal P(s,m,\ell)$. The two main uniformities to understand the construction are $m$ and $m+1$. One sees that the family contains all sets of size $m+1$, as well as all sets of size $m$ that intersect $[\ell-1]$, i.e., the family $\aaa_1^{(m)}(n,\ell-1)$. Thus, in fact, the sets of uniformity $m$ form a family with no matching of size $\ell$. We should note here that the sets in $\mathcal P(s,m,\ell)\cap {[n]\choose \le m}$, a bulk of which is $\aaa_1^{(m)}(n,\ell-1)$, could be replaced by any other family in ${[n]\choose m}$ with no matching of size $\ell$. As a result, we would obtain a construction of a family in $2^{[n]}$ with no $s$-matching.

In order to state the result, we describe the prospective extremal families. We have already seen the first example.
$$\mathcal{P}(s, \ell) := \mathcal P(s,2,\ell)=\big\{P \in 2^{[n]} : |P| + |P \cap [\ell - 1]| \geq 3\big\}.$$
The second family also contains all sets of size at least $3$, but differs on the $2$-uniform layer. It has the other extremal example for $2$-uniform families avoiding an $\ell$-matching: a clique on $2\ell-1$ vertices.
$$\mathcal{P}'(s, \ell) := {[n] \choose \geq 3} \cup {[2\ell-1] \choose 2}.$$
The third family misses some $3$-element sets, and instead has a larger $2$-uniform layer. It is also a clique. $$\mathcal{Q}(s, \ell) := {[n] \choose \geq 3} \cup {[s+\ell-1] \choose 2} \setminus {[s+\ell, n] \choose 3}.$$
Finally, the fourth family is an iterated doubling of the extremal family for $n=2s-1$.
$$\mathcal{W}(s, \ell) := \big\{P \in 2^{[n]} : |P \cap [2s - 1]| \geq 2\big\}.$$ We note that none of these families contain $0$- or $1$-element sets. In the next section, we show that all these families indeed avoid $s$-matchings.

The main result of the paper is the following theorem.

The main result of the paper is the following theorem.

\begin{figure}

\begin{theorem} \label{t.truncated_lattice}
 Let $n,s,\ell,c$ be positive integers such that $n=2s+c=3s-\ell$, and $c,\ell\in [s-1]$. If $\mathcal{F} \subset {[n] \choose \leq 3}$ has no $s$-matching, then
 $$|\mathcal{F}| \leq \max\big\{|\mathcal{P}(s, \ell)^{(\leq 3)}|, |\mathcal{P}'(s, \ell)^{(\leq 3)}|, |\mathcal{Q}(s, \ell)^{(\leq 3)}|, |\mathcal{W}(s, \ell)^{(\leq 3)}|\big\}.$$
\end{theorem}

\begin{theorem} \label{t.Erdos_Gallai}
 Let $n,s$ be positive integers, $n\geq 2s$. Suppose that $\mathcal{F} \subset 2^{[n]}$ satisfies $\nu(\mathcal{F}^{(2)}) < s$. Then $y(2) \geq \min({n - s + 1 \choose 2}, {n \choose 2} - {2s-1 \choose 2})$.
\end{theorem}
\begin{proof}
 As we have mentioned, we may assume that $\ff$ is shifted, cf.~\cite{F3}.
 Consider the sets $\{i, 2s+1-i\}$, $i\in [s]$. These sets form an $s$-matching, and thus at least one of them is not in $\mathcal{F}$. By Lemma \ref{l.EG}, we get $$y(2) \geq \frac{(n+2s+1-3i)(n-2s+i)}{2}$$ for some $i \in [1, s]$. The right-hand side is upward convex as a function of $i$. Thus, its minimum in the interval $[1, s]$ is attained at one of its ends. We get the bound
 $$y(2) \geq \min\Big\{{n - s + 1 \choose 2}, \frac{(n+2s-2)(n-2s+1)}{2}\Big\}.$$ This is the claimed bound since ${n \choose 2} - {2s-1 \choose 2} = \frac{(n+2s-2)(n-2s+1)}{2}$.
\end{proof}

\begin{thm}\label{t.relaxed} Let $n,s,\ell$ be non-negative integers such that $n = 3s-\ell$ and $0\le \ell\le s$. Suppose that $\mathcal{F} \subset 2^{[n]}$ is a shifted up-set with $\nu(\mathcal{F}) < s$ and $\mathcal{F} \cap {[n] \choose 1} = \emptyset$. Then $|\mathcal{F}| \leq \max\big\{|\mathcal{P}(s, \ell)|, |\mathcal{P}'(s, \ell)|, |\mathcal{Q}(s, \ell)|, |\mathcal{W}(s, \ell)|\big\}$. Moreover, equality is achieved only if $\mathcal{F}$ coincides with one of these families.
\end{thm}
\begin{proof}[Proof of Theorem~\ref{t.main} using Theorem~\ref{t.relaxed}]
We prove Theorem~\ref{t.main} by induction on $\ell$. For $\ell \in \{0, 1\}$ the statement is given by Theorem~\ref{thmkl}, together with uniqueness that is proven in \cite{Kl}. In what follows, we assume that $\ell\ge 2.$

If $\{1\} \notin \mathcal{F}$ then by shiftedness $\mathcal{F} \cap {[n] \choose 1} = \emptyset$ and we are done.
If $\{1\} \in \mathcal{F}$ then we apply induction. Let $\mathcal{F'} = \mathcal{F} \cap 2^{[2, n]}$. The family $\m F'$ is a family of subsets of an $(n-1)$-element set, $\nu(\mathcal{F'}) < s-1$, and we have $n-1 =3(s-1)-(\ell-2).$ Note that $\ell-2\ge 0$ and $\ell-2<s-1$, and thus we may apply induction to $\m F'$, getting that
$$|\overline{\mathcal{F'}}| \geq \min\big\{|\overline{\mathcal{P'}(s-1, \ell-2)}|, |\overline{\mathcal{P}(s-1, \ell-2)}|, |\overline{\mathcal{Q}(s-1, \ell-2)}|, |\overline{\mathcal{W}(s-1, \ell-2)}|\big\},$$
where the complement is taken with respect to $2^{[2,n]}.$ Clearly, $|\overline{\mathcal{F}}|\ge |\overline{\mathcal{F'}}|$, and so Theorem~\ref{t.main} is implied by the following claim.
\begin{restatable}{cla}{claimeleven} We have
 \begin{multline*}
 \min\big\{|\overline{\mathcal{P'}(s-1, \ell-2)}|, |\overline{\mathcal{P}(s-1, \ell-2)}|, |\overline{\mathcal{Q}(s-1, \ell-2)}|, \\|\overline{\mathcal{W}(s-1, \ell-2)}|\big\} >\min\big\{|\overline{\mathcal{P'}(s, \ell)}|, |\overline{\mathcal{P}(s, \ell)}|, |\overline{\mathcal{Q}(s, \ell)}|, |\overline{\mathcal{W}(s, \ell)}|\big\}.
 \end{multline*}
 Moreover,
 \begin{multline*}
 \min\big\{|\overline{\mathcal{P'}(s-1, \ell-2)}^{(\leq 3)}|, |\overline{\mathcal{P}(s-1, \ell-2)}^{(\leq 3)}|, |\overline{\mathcal{Q}(s-1, \ell-2)}^{(\leq 3)}|, \\|\overline{\mathcal{W}(s-1, \ell-2)}^{(\leq 3)}|\big\} >\min\big\{|\overline{\mathcal{P'}(s, \ell)}^{(\leq 3)}|, |\overline{\mathcal{P}(s, \ell)}^{(\leq 3)}|, |\overline{\mathcal{Q}(s, \ell)}^{(\leq 3)}|, |\overline{\mathcal{W}(s, \ell)}^{(\leq 3)}|\big\}.
 \end{multline*}
\label{c.no_siggletons}
\end{restatable}

The case of odd $d$ requires a more careful analysis. In this case, we use the inequality \eqref{eqd2c} which states that $d(\mathcal{F}) \leq 2c$.
\begin{lemma} \label{l.odd_d_to_y2}
 Let $d$ be a positive odd integer, $d \leq 2c$. If $\mathcal{F} \subset 2^{[n]}$ is a shifted family with $d(\mathcal{F}) = d$, then {\small \begin{equation}\label{eqy22}y(2) \geq \min\Big\{\frac{(4\ell+3c+d-2)(3c-d+1)}{2}, \frac{(\ell+3c-\frac{d-1}{2})(\ell+3c-\frac{d+1}2)}{2}\Big\}.\end{equation}} Moreover, equality is achieved only if $\mathcal{F}^{(2)} = {[2\ell+d-1] \choose 2}$ or $\mathcal{F}^{(2)} = \{F \in {n \choose 2 }: F\cap[\ell+\frac{d-1}{2}] \neq \emptyset\}$.
\end{lemma}

\subsection{$c \in \{3, 4\}$}
\begin{lemma} \label{l.c_eq_3}
 Let $n = 2s+3$ and $\mathcal{F} \subset 2^{[n]}$ is a shifted up-set with $\nu(\mathcal{F}) < s$ and $\mathcal{F} \cap {[n] \choose 1} = \emptyset$. Then $|\mathcal{F}| \leq \max(|\mathcal{P}(s, \ell)|, |\mathcal{P}'(s, \ell)|, |\mathcal{Q}(s, \ell)|)$. Moreover, equality is achieved only if $\mathcal{F}$ is one of the families $\mathcal{P}(s, \ell), \mathcal{P}'(s, \ell), \mathcal{Q}(s, \ell)$.
\end{lemma}

\begin{figure}

\centering

\includegraphics[width=0.8\linewidth]{extremal_families.png}

\caption{Extremal families. $\mathcal{W}$ is one of the extremal families for $c = 1, s \geq 5$.}

\label{fig1}

\end{figure}

\begin{figure}

\centering

\includegraphics[width=0.6\linewidth]{extremal_families_2.png}

\caption{Extremal families for small $c,s$. Note that for $c=1, s=5$ there are $3$ different extremal families. For $\ell = 1$, that is, $s = c + 1$, the family $\mathcal{P}'$ coincides with $\mathcal{P}$ (both are equal to ${n \choose \geq (m + 1)}$) and thus is formally extremal. When $\mathcal{P}' \neq \mathcal{P}$, for $c \leq 9$ $\mathcal{P}'$ cannot be extremal.}

\label{fig2}

\end{figure}
On Figures~\ref{fig1} and~\ref{fig2} we show, which families are extremal for different regimes of the parameters $s,c$. For some values we get that three different families are extremal at the same time.

We define shifted families in the next section. We should note that actually there are rather natural examples of families with no $s$-matching interpolating between $\mathcal{P}'(s, l)$ and $\mathcal{Q}(s, l)$ in a somewhat similar way as $\aaa_i$ interpolate between $\aaa_0$ and $\aaa_k$, but, as in the case of the EMC, there is a certain convexity that leads to the fact that it is the endpoints that must be extremal.

In the proof we will work only with sets of size $3$ or less. Therefore, any family, avoiding $s$-matching, must miss at least as many sets of size $3$ or less, as the extremal family. We thus get the following theorem about the truncated boolean lattice, confirming a conjecture of Frankl and the first author \cite{FK9} in our regime of the parameters.

\begin{theorem} \label{t.truncated_lattice}
    Let $n,s,\ell,c$ be positive integers such that $n=2s+c=3s-\ell$, and $c,\ell\in [s-1]$. If $\mathcal{F} \subset {[n] \choose \leq 3}$ has no $s$-matching, then
    $$|\mathcal{F}| \leq \max\big\{|\mathcal{P}(s, \ell)^{(\leq 3)}|, |\mathcal{P}'(s, \ell)^{(\leq 3)}|, |\mathcal{Q}(s, \ell)^{(\leq 3)}|, |\mathcal{W}(s, \ell)^{(\leq 3)}|\big\}.$$
\end{theorem}

Note that a similar statement about $2$ first layers of boolean lattice is obviously false. Indeed, one of the families $\aaa_1^{(2)}(n,s-1), \aaa_2^{(2)}(n,s-1)$ has a larger cardinality than families $\mathcal{P}(s, \ell)^{(\leq 2)}, \mathcal{P}'(s, \ell)^{(\leq 2)}, \mathcal{Q}(s, \ell)^{(\leq 2)}, \mathcal{W}(s, \ell)^{(\leq 2)}$.

In Section~\ref{sec2}, we prove several easy facts and make the necessary preparations for the proof of the main theorem. In Section~\ref{sec3} we describe the strategy of the proof of the main theorem.

To prove Theorem~\ref{t.main}, the authors note that they will "work only with sets of size $3$ or less." Hence, "any family, avoiding $s$-matching, must miss at least as many sets of size $3$ or less, as the extremal family." This reduction leads to an auxiliary result on the truncated Boolean lattice: Theorem~\ref{t.truncated_lattice}, which bounds $|\mathcal F\subset {[n]\choose \le 3}|$ with no $s$-matching by the maximum of the truncated versions of the candidate extremal families $\mathcal P,\mathcal P',\mathcal Q,\mathcal W$. They also mention that while there are "natural examples" interpolating between $\mathcal P'$ and $\mathcal Q$, "there is a certain convexity" implying "it is the endpoints that must be extremal." Finally, they indicate the paper structure: in Section~\ref{sec2} they "prove several easy facts and make the necessary preparations" and in Section~\ref{sec3} they "describe the strategy of the proof of the main theorem."

To prove the main theorem, the authors note that they will "work only with sets of size $3$ or less." Hence, "any family, avoiding $s$-matching, must miss at least as many sets of size $3$ or less, as the extremal family." This reduction leads to an auxiliary result on the truncated Boolean lattice: Theorem~\ref{t.truncated_lattice}, which bounds $|\mathcal F\subset {[n]\choose \le 3}|$ with no $s$-matching by the maximum of the truncated versions of the candidate extremal families $\mathcal P,\mathcal P',\mathcal Q,\mathcal W$. They also mention that while there are "natural examples" interpolating between $\mathcal P'$ and $\mathcal Q$, "there is a certain convexity" implying "it is the endpoints that must be extremal." Finally, they indicate the paper structure: next they "prove several easy facts and make the necessary preparations" and later they "describe the strategy of the proof of the main theorem."

Classification or Bijection
Equality or Bound

Question:
Let \([n]=\{1,\dots,n\}\), and define
\[
e(n,s)=\max\{\,|\mathcal F|: \mathcal F\subset 2^{[n]},\ \nu(\mathcal F)<s\,\},
\]
where \(\nu(\mathcal F)\) is the largest size of a pairwise disjoint subfamily of \(\mathcal F\). A family \(\mathcal F\subset 2^{[n]}\) is called shifted if whenever \(A\in\mathcal F\), \(j\in A\), \(i\notin A\), and \(1\le i<j\le n\), then \((A\setminus\{j\})\cup\{i\}\in\mathcal F\). For positive integers \(n,s,\ell,c\) satisfying \(n=2s+c=3s-\ell\) and \(c,\ell\in\{1,\dots,s-1\}\), let
\[
\mathcal P(s,\ell)=\{A\subset [n]: |A|+|A\cap[\ell-1]|\ge 3\},
\]
and let \(\mathcal P'(s,\ell),\mathcal Q(s,\ell),\mathcal W(s,\ell)\) be the other three specified candidate families. Which statement gives the correct extremal description?

One has
\[
e(n,s)=\max\big\{|\mathcal P(s,\ell)|,\ |\mathcal P'(s,\ell)|,\ |\mathcal Q(s,\ell)|,\ |\mathcal W(s,\ell)|\big\}.
\]
Moreover, if \(s\ge 3\), \(\mathcal F\subset 2^{[n]}\) is shifted, \(\nu(\mathcal F)<s\), and \(|\mathcal F|=e(n,s)\), then \(\mathcal F\) is exactly one of \(\mathcal P(s,\ell),\mathcal P'(s,\ell),\mathcal Q(s,\ell),\mathcal W(s,\ell)\).

Choices:
B.
One has
\[
e(n,s)=\max\big\{|\mathcal P(s,\ell)|,\ |\mathcal P'(s,\ell)|,\ |\mathcal Q(s,\ell)|,\ |\mathcal W(s,\ell)|\big\}.
\]
Moreover, if \(s\ge 3\), \(\mathcal F\subset 2^{[n]}\) is shifted, \(\nu(\mathcal F)<s\), and \(|\mathcal F|=e(n,s)\), then \(\mathcal F\) is exactly one of \(\mathcal P(s,\ell),\mathcal P'(s,\ell),\mathcal Q(s,\ell),\mathcal W(s,\ell)\), and this extremal family is unique for the given parameters \((n,s,\ell,c)\).
C.
One has
\[
e(n,s)=\max\big\{|\mathcal P(s,\ell)|,\ |\mathcal P'(s,\ell)|,\ |\mathcal Q(s,\ell)|,\ |\mathcal W(s,\ell)|\big\}.
\]
D.
One has
\[
e(n,s)=\max\big\{|\mathcal P(s,\ell)|,\ |\mathcal P'(s,\ell)|,\ |\mathcal Q(s,\ell)|,\ |\mathcal W(s,\ell)|\big\}.
\]
Moreover, if \(s\ge 3\), \(\mathcal F\subset 2^{[n]}\) satisfies \(\nu(\mathcal F)<s\) and \(|\mathcal F|=e(n,s)\), then \(\mathcal F\) is exactly one of \(\mathcal P(s,\ell),\mathcal P'(s,\ell),\mathcal Q(s,\ell),\mathcal W(s,\ell)\).
E.
One has
\[
e(n,s)=\max\big\{|\mathcal P(s,\ell)|,\ |\mathcal P'(s,\ell)|,\ |\mathcal Q(s,\ell)|,\ |\mathcal W(s,\ell)|\big\}.
\]
Moreover, if \(s\ge 3\), \(\mathcal F\subset 2^{[n]}\) is shifted, \(\nu(\mathcal F)<s\), and \(|\mathcal F|=e(n,s)\), then \(\mathcal F\) is exactly one of the truncated families \(\mathcal P(s,\ell)\cap {[n]\choose \le 3},\ \mathcal P'(s,\ell)\cap {[n]\choose \le 3},\ \mathcal Q(s,\ell)\cap {[n]\choose \le 3},\ \mathcal W(s,\ell)\cap {[n]\choose \le 3}\).

multiple-endpoint extremals vs uniqueness

dropped shifted-extremal classification clause

shifted hypothesis in uniqueness statement

truncated-lattice reduction promoted to full classification

The stem does not state the correct conclusion outright and does not single out choice A. It only sets up the objects and asks for the correct extremal description.

This is close to theorem-statement recognition: the correct option is essentially the exact extremal theorem, with distractors formed by slight logical modifications. It is not fully tautological, but it is only a mild reformulation of a known result.

Some reasoning is needed to distinguish the exact classification from stronger, weaker, or hypothesis-altered variants, especially around uniqueness and the shifted assumption. However, it mainly tests precise recall/statement matching rather than deeper derivation.

The distractors are plausible and mathematically meaningful: one is a weaker true statement, one overstrengthens to uniqueness, one drops a crucial hypothesis, and one introduces an unjustified truncation. These reflect realistic failure modes.

A solid theorem-recognition MCQ with strong distractors and little answer leakage, but it leans more toward precise statement identification than genuinely generative mathematical reasoning.


\label{MainTheorem}
    Let $g \in L^2(\mathbb{R})$ with Fourier transform satisfies the super-exponential decay condition:
    \begin{equation*}
        |\hat{g}(\xi)| \leq C e^{-a|\xi|^m}
    \end{equation*}
    for some constants $m>1$, $C>0$, $a \in (1,\infty)$. Let $\tau_1, \tau_2 \in \mathbb{R}^+$ satisfy
    \begin{equation*}
        \tau_1 < \big(\frac{2}{(2\pi)^{m/(m-1)}(ma)^{-1/(m-1)}e}\big)^{\frac{m-1}{m}}, \quad \quad \quad \tau_2< (\frac{2}{ame})^{\frac{1}{m}}.
    \end{equation*}
    Define the set $\Lambda$ such that:
    \begin{equation*}
        \Lambda =  \{(\pm\tau_1n^{\frac{m-1}{m}},\pm\tau_2n^{\frac{1}{m}}),n \in \mathbb{N}\}
    \end{equation*}
    Then the following statements are equivalent for every $f, h \in L^2(\mathbb{R}):$
    \begin{enumerate}
        \item $|V_g(f)(\lambda)| = |V_gh(\lambda)|$ for every $\lambda \in \Lambda$.
        \item $f = e^{i \alpha}h$ for some $\alpha \in [0,2\pi).$
    \end{enumerate}

\begin{theorem}\label{MainTheorem}
    Let $g \in L^2(\R)$ with Fourier transform satisfies the super-exponential decay condition:
    \begin{equation*}
        |\hat{g}(\xi)| \leq C e^{-a|\xi|^m}
    \end{equation*}
    for some constants $m>1$, $C>0$, $a \in (1,\infty)$. Let $\tau_1, \tau_2 \in \R^+$ satisfy
    \begin{equation*}
        \tau_1 < \big(\frac{2}{(2\pi)^{m/(m-1)}(ma)^{-1/(m-1)}e}\big)^{\frac{m-1}{m}}, \quad \quad \quad \tau_2< (\frac{2}{ame})^{\frac{1}{m}}.
    \end{equation*}
    Define the set $\Lambda$ such that:
    \begin{equation*}
        \Lambda =  \{(\pm\tau_1n^{\frac{m-1}{m}},\pm\tau_2n^{\frac{1}{m}}),n \in \N\}
    \end{equation*}
    Then the following statements are equivalent for every $f, h \in L^2(\R):$
    \begin{enumerate}
        \item $|V_g(f)(\lambda)| = |V_gh(\lambda)|$ for every $\lambda \in \Lambda$.
        \item $f = e^{i \alpha}h$ for some $\alpha \in [0,2\pi).$
    \end{enumerate}
\end{theorem}

Context:
\label{sec:1}
\input{introduction}

\label{sec:1}
\input{introduction}

\section{Preliminary}
\label{sec:2}
A central principle in Fourier analysis is the connection between the smoothness of a function and the decay rate of its Fourier transform at infinity. The Paley-Wiener theorem and its variants are the primary tools that precisely characterize this relationship, providing the foundation for extending functions into the complex domain based on the behavior of their Fourier transforms.
\subsection{Paley-Wiener Type Theorem}
The Fourier transform of a Lebesgue integrable function $f$ on $\R$ is given by
\begin{equation*}
 \mathcal{F}(f):=\hat{f} (\xi) = \int_{\R} f(t) e^{-2 \pi i t \cdot\xi} dt, \quad \xi \in \R.
\end{equation*}

A standard density argument can be used to extend this definition so that $\mathcal{F}$ is an isometry on $L^2(\R)$. The Fourier transform of a function $f \in L^2(\R)$ is said to be of \emph{super-exponential decay} if
\begin{equation}\label{FourierDecay}
 |\hat{f}(\xi)| \lesssim Ce^{-a|\xi|^m}
\end{equation}
for some $m>1$, $C>0$ and $a \in (1,\infty)$. In this subsection we introduce a slightly modified version of Paley-Wiener theorem. In particular, we show that if the Fourier transform of a function $f$ decays super-exponential, then $f$ can
be extended to an entire function whose order and type depend on the parameters $a$ and $m$. To establish this result, we make use of Morera’s Theorem, which we now recall.
\begin{mainthm}\cite[Theorem 5.2]{stein2010complex}\label{Morera}
 If $\{ f_n \}_{n \in \N}$ is a sequence of holomorphic functions that converges uniformly to a function $f$ in every compact subset of $\Omega$, then $f$ is holomorphic in $\Omega$.
\end{mainthm}

Similarly, fix $z=x+iy$ and consider $z' = x'+iy'\subset K'$, where $K'$ is compact. Then we have
\begin{equation}\label{fixz}
 \int_{-\infty}^{\infty} |\overline{g(t-z)}|^2 |e^{2\pi i z' \xi}|^2 dt \leq C\int_{-\infty}^{\infty} e^{-2a|t-x|^{\rho}+2b|y|^{\rho}-4\pi y't}dt.
\end{equation}
Since $z=x+iy$ is fixed, for $z' \in K'$ it follows that \eqref{fixz} converges for $z' \in K$. Following a statement from \cite[Page 68]{boas2011entire}, the order and type of an entire function is retained under integral, therefore the orders and types for $|V_gf (z,z')|$ can be computed directly using Lemma \ref{OrderType}. It is noteworthy that our computation for orders is compatible with the result from \cite[Lemma 3.4]{grohs2024phaseless} when $m = 2$.
\end{proof}
\section{Main Result}
Denote the ring of holomorphic functions by $\mathcal{O}(\C)$. Consider the collection of entire functions with $\rho>1$ and $a,b>0$:
$$\fG_{a,b,\rho} = \{ f \in \mathcal{O}(\C)\big| |f(x+iy)| \lesssim e^{-a|x|^{\rho}+b|y|^{\rho}} \}.$$
It is straightforward to see that $\fG_{a,b,\rho}$ is a linear space. The objective is to establish a uniqueness set so that whenever two functions $f, g \in \fG_{a,b,\rho}$ agree on the set, they agree \textit{everywhere}. Recent literature shows that such functions cannot be uniquely determined from their values on integer lattices \cite{alaifari2021phase}. For the purpose of notation, let $U \subset \mathcal{O(\C)}$ denote a linear function space. A set $\Lambda \subset \C$ is called a uniqueness set of $U$ if for any $f \in U$, it holds that
$$(f( \lambda ) = 0 \; \text{for all} \; \lambda \in \Lambda) \Longrightarrow f = 0 \; \text{everywhere on } \: \C.$$
Since zero of any entire function can't be an accumulation point, any open set in $\C$ is trivially a uniqueness set. Without this property, determining discrete uniqueness sets of $\fG_{a,b,\rho}$ contained in $\R$ is a nontrivial task. We first propose a sufficient condition for such a set.\\

\input{references}
\end{document}
OrderType.tex0000664000000000000000000001265715107436001012225 0ustar rootroot
\begin{lemma}\label{OrderType}
Let $f \in L^2(\mathbb{R})$ be a function whose Fourier transform satisfies $|\hat{f}(\xi)| \le C e^{-a|\xi|^m}$ for some $m>1$ and $a>0$. Then its analytic continuation $f(z)$ is an entire function of type at most $\tau = \frac{m-1}{m}(2\pi)^{m/(m-1)}(am)^{-1/(m-1)}$ corresponding to maximum possible order $\rho = \frac{m}{m-1}$.
\end{lemma}
\begin{proof}
The analytic continuation of the function, $f(z)$, is given by the inverse Fourier transform:
\begin{equation*}
 f(z) = \int_{-\infty}^\infty \hat{f}(\xi) e^{2\pi i z \xi} d\xi
\end{equation*}
The Taylor coefficients of $f(z)$ are given by
\begin{equation*}
c_n = \frac{f^{(n)}(0)}{n!} = \frac{(2\pi i)^n}{n!} \int_{-\infty}^\infty \xi^n \hat{f}(\xi) d\xi
\end{equation*}
We can now bound the magnitude of the coefficients using the given decay of $\hat{f}(\xi)$:
\begin{equation*}
|c_n| \le \frac{(2\pi)^n}{n!} \int_{-\infty}^\infty |\xi|^n |\hat{f}(\xi)| d\xi \le \frac{C(2\pi)^n}{n!} \int_{-\infty}^\infty |\xi|^n e^{-a|\xi|^m} d\xi
\end{equation*}
The integral can be evaluated using the Gamma function:
\begin{equation*}
 \int_{-\infty}^\infty |\xi|^n e^{-a|\xi|^m} d\xi = \frac{2}{m a^{(n+1)/m}} \Gamma\left(\frac{n+1}{m}\right).
\end{equation*}
Using Stirling's formula for Gamma function, we obtain
\begin{equation*}
 \Gamma(\frac{n+1}{m}) = \sqrt{\frac{2 \pi m}{n+1}}(\frac{n+1}{me})^{(n+1)/m}\big(1+ O(\frac{m}{n+1})\big).
\end{equation*}
and
\begin{equation*}
 n! \approx \sqrt{2 \pi n}(\frac{n}{e})^n
\end{equation*}
Then, for some $C' >0$, we derive the following expression:
 \begin{align}\label{GeneralTaylorCoefficient}
 |c_n|
 &\lesssim \Big((2\pi)^n \cdot n^{-\frac{1}{2}-n}\cdot e^{n -\frac{n+1}{m}}\cdot a^{-\frac{n+1}{m}}\cdot m ^{-\frac{1}{2}-\frac{n+1}{m}}\cdot (n+1)^{\frac{n+1}{m}-\frac{1}{2}}\Big)\cdot\Big(1 + \frac{C'm}{n+1}\Big) \\
 &= R(a,n,m)\cdot n^{-n}\cdot(n+1)^{\frac{n+1}{m}}\cdot\Big(1 + \frac{C'm}{n+1}\Big) \nonumber
 \end{align}
The term $R(a,n,m)$ collects all remaining exponential and polynomial factors. It follows that:
\begin{align*}
 \rho &= \limsup_{n\to\infty} \frac{n \log n}{-\log|c_n|} \lesssim \limsup_{n\to\infty} \frac{n\log n}{-\log R(a,m,n) + n\log n -\frac{n+1}{m}\log(n+1)- \log(1+\frac{C'm}{n+1})}\\
 &\lesssim \limsup_{n\to\infty} \frac{1}{\frac{-\log R(a,m,n)}{n\log n} + 1 -\frac{n+1}{nm}\frac{\log(n+1)}{\log n}- \frac{\log(1+\frac{C'm}{n+1})}{n\log n}}.
\end{align*}
A series of computations shows that $$\begin{cases} \lim_{n\to \infty}\frac{\log R(a,m,n)}{n\log n} =0\\ \lim_{n\to \infty}\frac{\log(1+ \frac{C'm}{n+1})}{n\log n}=0\end{cases}.$$ Therefore, the remaining terms are:
\begin{equation*}
 \rho \leq \limsup_{n\to \infty}\frac{1}{1-\frac{n+1}{mn}\frac{\log(n+1)}{\log n}}= \frac{1}{1- \frac{1}{m}} = \frac{m}{m-1}.
\end{equation*}
Let $\tilde{\rho} = \frac{m}{m-1}$, and choose $A,R >0$ such that
\begin{equation}\label{TypeWRTorder}
 \max_{|z|= r}|f(z)| < Re^{Ar^{\tilde{\rho}}}.
\end{equation}
Now we compute the type $\tau$ of $f(z)$ with respect to order $\tilde{\rho}$, i.e., the greatest lower bound of $A$ in Equation~\eqref{TypeWRTorder}. We note that

\begin{figure}[h!]
    \centering
    \includegraphics[width=0.7\textwidth]{rhosampling.png}
    \caption{An illustration of sampling points $\Lambda$ in time-frequency plane for $g$ with Fourier decay parameter $m=\frac{3}{2}$ and $a=1$}
\end{figure}
{The rest of the paper is organized as follows: In Section 2 we introduce some necessary preliminaries. Subsequently we prove Theorem~\ref{MainTheorem} in Section 3. To this end, we first establish a sufficient condition for a discrete set to be a uniqueness set for the relevant class of entire functions (Proposition 1) and use Proposition 2 to show that our result is in a sense "sharp". We then use these technical results to prove Theorem~\ref{MainTheorem}, which provides a concrete discrete sampling set $\Lambda$ that guarantees the unique recovery of $f$ up to a global phase.

To prove Theorem~\ref{MainTheorem}, the paper (in Section 3) proceeds by first establishing “a sufficient condition for a discrete set to be a uniqueness set for the relevant class of entire functions (Proposition 1),” then using “Proposition 2 to show that our result is in a sense sharp,” and finally applying “these technical results to prove Theorem~\ref{MainTheorem},” yielding “a concrete discrete sampling set $\Lambda$ that guarantees the unique recovery of $f$ up to a global phase.”

To prove the main theorem, the paper proceeds by first establishing a sufficient condition for a discrete set to be a uniqueness set for the relevant class of entire functions (Proposition 1), then using Proposition 2 to show that the result is in a sense sharp, and finally applying these technical results to prove the main theorem, yielding a concrete discrete sampling set $\Lambda$ that guarantees the unique recovery of $f$ up to a global phase.

Biconditional or Equivalence
Classification or Bijection

Question:
Let the short-time Fourier transform with window \(g\) be \(V_g f(x,\xi)=\int_{\mathbb R} f(t)\overline{g(t-x)}e^{-2\pi i\xi t}\,dt\). Suppose \(g\in L^2(\mathbb R)\) has Fourier transform satisfying
\[
|\widehat g(\xi)|\le C e^{-a|\xi|^m}
\]
for some \(m>1\), \(C>0\), and \(a>1\). Let \(\tau_1,\tau_2>0\) satisfy
\[
\tau_1<\Big(\frac{2}{(2\pi)^{m/(m-1)}(ma)^{-1/(m-1)}e}\Big)^{\frac{m-1}{m}},
\qquad
\tau_2<\Big(\frac{2}{ame}\Big)^{1/m},
\]
and define
\[
\Lambda=\{(\pm \tau_1 n^{(m-1)/m},\, \pm \tau_2 n^{1/m}) : n\in\mathbb N\}\subset \mathbb R^2,
\]
with all sign choices allowed. Which statement exactly characterizes the pairs \(f,h\in L^2(\mathbb R)\) for which
\(|V_g f(\lambda)|=|V_g h(\lambda)|\) for every \(\lambda\in\Lambda\)?

They are exactly the pairs differing by a global phase: \(|V_g f(\lambda)|=|V_g h(\lambda)|\) for all \(\lambda\in\Lambda\) if and only if \(f=e^{i\alpha}h\) for some \(\alpha\in[0,2\pi)\).

Choices:
B. They are exactly the pairs related by a time-frequency shift and a global phase: \(|V_g f(\lambda)|=|V_g h(\lambda)|\) for all \(\lambda\in\Lambda\) if and only if there exist \(x_0,\xi_0\in\mathbb R\) and \(\alpha\in[0,2\pi)\) such that \(f(t)=e^{i\alpha}e^{2\pi i \xi_0 t}h(t-x_0)\) a.e.
C. If \(f=e^{i\alpha}h\) for some \(\alpha\in[0,2\pi)\), then \(|V_g f(\lambda)|=|V_g h(\lambda)|\) for every \(\lambda\in\Lambda\).
D. They are exactly the pairs differing by a global phase provided the equalities hold on one fixed-sign branch of the sampling set, i.e. if and only if \(|V_g f(\tau_1 n^{(m-1)/m},\tau_2 n^{1/m})|=|V_g h(\tau_1 n^{(m-1)/m},\tau_2 n^{1/m})|\) for every \(n\in\mathbb N\), then \(f=e^{i\alpha}h\) for some \(\alpha\in[0,2\pi)\).
E. They are exactly the pairs differing by a global phase for any choice of \(\tau_1,\tau_2>0\): \(|V_g f(\lambda)|=|V_g h(\lambda)|\) for all \(\lambda\in\Lambda\) if and only if \(f=e^{i\alpha}h\) for some \(\alpha\in[0,2\pi)\), without requiring the upper bounds on \(\tau_1\) and \(\tau_2\).

The stem does not state or strongly hint at the correct conclusion; it only gives the hypotheses and asks for the exact characterization.

This is close to a theorem-recall item: the stem essentially presents the assumptions and asks for the precise conclusion. It is not a pure tautology because the options vary in quantifiers, hypotheses, and equivalence strength.

Some reasoning is needed to distinguish the exact iff statement from weaker, stronger, or hypothesis-altered variants, but the task is mainly precise theorem recognition rather than substantial derivation.

The distractors are plausible and target real failure modes: confusing global phase with time-frequency symmetry, accepting only one implication, dropping the four-sign sampling requirement, or removing the tau bounds.

A solid theorem-identification MCQ with strong distractors and no answer leakage, though it leans more toward exact statement recall than deeper generative reasoning.


\label{thm:intro-keller}
    Let $X$ be a quasi-compact separated scheme over a base field $\mathbb{K}$. Then there exist canonical isomorphisms
    \begin{align*}
        & HH_\bullet\big(\Perf(X)\big) \to HH_\bullet(X),\\
        & HC_\bullet\big(\Perf(X)\big) \to HC_\bullet(X).
    \end{align*}

\begin{thm}\label{thm:intro-keller}
    Let $X$ be a quasi-compact separated scheme over a base field $\bbK$. Then there exist canonical isomorphisms
    \begin{align*}
        & HH_\bullet\big(\Perf(X)\big) \to HH_\bullet(X),\\
        & HC_\bullet\big(\Perf(X)\big) \to HC_\bullet(X).
    \end{align*}
\end{thm}

Let $X$ be a quasi-compact separated scheme over a base field $\mathbb{K}$. An important set of invariants associated with $X$ is its Hochschild homology $HH_\bullet(X)$ and cyclic homology $HC_\bullet(X)$. They are defined as the hyper-cohomology groups of the sheafification of the Hochschild complexes and cyclic chain complexes respectively.

On the other hand, let $\Perf(X)$ denote the dg category of perfect complexes over $X$. We emphasize that this is a dg-enhancement of the full subcategory of the bounded derived category $D^b(X)$ consisting of perfect complexes over $X$. For any small $\mathbb{K}$-linear dg category, we may also define purely algebraically its own Hochschild type invariants. In the case of $\Perf(X)$, we are naturally lead to the question if $HH_\bullet\big(\Perf(X)\big)$ (respectively $HC_\bullet\big(\Perf(X)\big)$) would be naturally isomorphic to the geometrically defined invariants $HH_\bullet(X)$ (respectively $HC_\bullet(X)$). In~\cite{keller1998cyclic}, Keller obtained the following result.

\begin{abstract}
Let $X$ be a quasi-compact separated scheme over a base field. Keller proved a theorem stating that the cyclic homology of $X$ is canonically isomorphic to the cyclic homology of the dg category $\Perf(X)$ consisting of perfect complexes over $X$. This theorem shows the categorical nature of the cyclic homology. In this note, we generalize Keller's theorem to allow $X$ be defined over a base commutative ring.
\end{abstract}

On the other hand, let $\Perf(X)$ denote the dg category of perfect complexes over $X$. We emphasize that this is a dg-enhancement of the full subcategory of the bounded derived category $D^b(X)$ consisting of perfect complexes over $X$. For any small $\bbK$-linear dg category, we may also define purely algebraically its own Hochschild type invariants. In the case of $\Perf(X)$, we are naturally lead to the question if $HH_\bullet\big(\Perf(X)\big)$ (respectively $HC_\bullet\big(\Perf(X)\big)$) would be naturally isomorphic to the geometrically defined invariants $HH_\bullet(X)$ (respectively $HC_\bullet(X)$). In~\cite{keller1998cyclic}, Keller obtained the following result.

In the study of logrithmic Hochschild homology~\cite{Olsson,HABLICSEK2026127}, one is often lead to consider Hochschild homology relative to a base space~\cite{OLSSON2003747}. This serves as at least one of the motivations to obtain a generalization of Keller's theorem above in the relative setting. More precisely, we prove the following

\begin{thm}
 Let $\mathfrak{X}$ be a quasi-compact separated scheme over a base commutative ring $R$. Then there are canonical isomorphisms
 \begin{align*}
 & HH_\bullet\big(\Perf(\mathfrak{X})/R\big) \to HH_\bullet(\mathfrak{X}/R),\\
 & HC_\bullet\big(\Perf(\mathfrak{X})/R\big) \to HC_\bullet(\mathfrak{X}/R).
 \end{align*}
\end{thm}

\subsection{Conventions and Notations.} We work over a base field $\mathbb{K}$. For a dg category $\CC$ over $\mathbb{K}$, denote by $\underline{\CC}$ its homotopy category. This is a category with the same underlying objects as $\CC$ and with
\[ {\sf Hom}_{\underline{\CC}}(X,Y):= H^0\big(\hom^\bullet_\CC(X,Y)\big).\]
For a scheme $X$, in Keller's original treatment~\cite{keller1998cyclic}, he uses the notion of localization pairs, i.e. a pair of dg categories $\big(\per(X),{\sf acyc}(X)\big)$ formed by the category of perfect complexes over $X$ and its full subcategory of acyclic perfect complexes. By a result of Drinfeld~\cite{drinfeld2004dg} on the construction of dg quotients of dg categories, we may first take the dg quotient $\per(X)/{\sf acyc}(X)$ as our definition of $\Perf(X)$. Due to this reason, in this paper we shall work exclusively with dg categories instead of localization pairs. In other words, in our setting, we have
\[ \underline{\Perf(X)} \cong \underline{\per(X)}/\underline{{\sf acyc}(X)}.\]
Thus no further localization is needed when working with $\Perf(X)$.

Let $X$ be a quasi-compact separated scheme over a base field $\bbK$. Keller considers a pair of dg categories $\big(\per(X),{\sf acyc}(X)\big)$ formed by the category of perfect complexes over $X$ and its full subcategory of acyclic perfect complexes. Then he defines the associated mixed complex as
\[ M\big(\per(X),{\sf acyc}(X)\big):={\sf cone}\big( M({\sf acyc}(X))\to M(\per(X))\big),\]
using the canonical inclusion functor. Throughout the paper, we shall denote the pair above by $\Perf(X):= \big(\per(X),{\sf acyc}(X)\big)$. By a result of Drinfeld~\cite{drinfeld2004dg} on the construction of dg quotients of dg categories, this choice of notation should cause no confusion. In other words, we may first take the dg quotient $\per(X)/{\sf acyc}(X)$ and then just take its associated mixed complex.

\begin{thm}\label{thm:main}
 Let $X$ be a quasi-compact separated scheme over a base field $\bbK$. Then trace map $\tau: M\big(\Perf(X)\big)\to R\Gamma(X,M\big(\cO)^\sharp)$ defined in Diagram~\eqref{diagram:tau} is invertible in $\DD Mix$.
\end{thm}

\begin{thm}\label{thm:main-relative}
 Let $\mathfrak{X}$ be a quasi-compact separated scheme over a base commutative ring $R$. Then trace map $\tau: M\big(\Perf(\mathfrak{X})\big)\to R\Gamma(\mathfrak{X},M\big(\cO)^\sharp)$ defined in Diagram~\eqref{diagram:tau} is invertible in $\DD Mix(R)$. Note that the mixed complex functor $M$ should be understood in the sense of Equation~\eqref{eq:mixed-ring}.
\end{thm}

In the study of logrithmic Hochschild homology~\cite{Olsson,HABLICSEK2026127}, one is often lead to consider Hochschild homology relative to a base space~\cite{OLSSON2003747}. This serves as at least one of the motivations to obtain a generalization of Keller's theorem above in the relative setting. More precisely, we prove the following

\begin{thm}
    Let $\mathfrak{X}$ be a quasi-compact separated scheme over a base commutative ring $R$. Then there are canonical isomorphisms
    \begin{align*}
        & HH_\bullet\big(\Perf(\mathfrak{X})/R\big) \to HH_\bullet(\mathfrak{X}/R),\\
        & HC_\bullet\big(\Perf(\mathfrak{X})/R\big) \to HC_\bullet(\mathfrak{X}/R).
    \end{align*}
\end{thm}

Note that in defining the Hochschild invariants above, when the structure sheaf $\cO_\mathfrak{X}$ is not flat over $R$, we need to replace it by a flat resolution. This homology theory is sometimes called Shukla-Hochschild homology~\cite{Shuk}. Similarly, such a replacement is also needed for the dg category $\Perf(\mathfrak{X})$. The proof of the relative case follows the same line as that of Theorem~\ref{thm:intro-keller}. Indeed, in Section~\ref{sec:review} we briefly sketch Keller's original proof. In Section~\ref{sec:relative} we deduce our main result by taking care of flatness using semi-free resolutions of dg categories.

\subsection{Conventions and Notations.} We work over a base field $\mathbb{K}$. For a dg category $\mathcal{C}$ over $\mathbb{K}$, denote by $\underline{\mathcal{C}}$ its homotopy category. This is a category with the same underlying objects as $\mathcal{C}$ and with
\[ {\sf Hom}_{\underline{\mathcal{C}}}(X,Y):= H^0\big(\hom^\bullet_\mathcal{C}(X,Y)\big).\]
For a scheme $X$, in Keller's original treatment~\cite{keller1998cyclic}, he uses the notion of localization pairs, i.e. a pair of dg categories $\big(\per(X),{\sf acyc}(X)\big)$ formed by the category of perfect complexes over $X$ and its full subcategory of acyclic perfect complexes. By a result of Drinfeld~\cite{drinfeld2004dg} on the construction of dg quotients of dg categories, we may first take the dg quotient $\per(X)/{\sf acyc}(X)$ as our definition of $\Perf(X)$. Due to this reason, in this paper we shall work exclusively with dg categories instead of localization pairs. In other words, in our setting, we have
\[ \underline{\Perf(X)} \cong \underline{\per(X)}/\underline{{\sf acyc}(X)}.\]
Thus no further localization is needed when working with $\Perf(X)$.

The post-theorem introduction does not sketch a proof of Theorem~\ref{thm:intro-keller} itself, but indicates the strategy for the relative generalization: it says that “The proof of the relative case follows the same line as that of Theorem~\ref{thm:intro-keller}.” It highlights the extra issue that “when the structure sheaf $\cO_\mathfrak{X}$ is not flat over $R$, we need to replace it by a flat resolution” (Shukla-Hochschild homology), and that “such a replacement is also needed for the dg category $\Perf(\mathfrak{X})$.” The plan of the argument is then: “in Section~\ref{sec:review} we briefly sketch Keller's original proof,” and “in Section~\ref{sec:relative} we deduce our main result by taking care of flatness using semi-free resolutions of dg categories.”

The post-theorem introduction does not sketch a proof of the main theorem itself, but indicates the strategy for the relative generalization: it says that “The proof of the relative case follows the same line as that of the main theorem.” It highlights the extra issue that “when the structure sheaf $\cO_\mathfrak{X}$ is not flat over $R$, we need to replace it by a flat resolution” (Shukla-Hochschild homology), and that “such a replacement is also needed for the dg category $\Perf(\mathfrak{X})$.” The plan of the argument is then: “next we briefly sketch Keller's original proof,” and “later we deduce our main result by taking care of flatness using semi-free resolutions of dg categories.”

Question: Let $X$ be a quasi-compact separated scheme over a base field $\mathbb{K}$, and let $\Perf(X)$ denote the dg category of perfect complexes on $X$. Which invariants are canonically isomorphic to the Hochschild homology $HH_\bullet(\Perf(X))$ and the cyclic homology $HC_\bullet(\Perf(X))$?

They are canonically isomorphic to the geometrically defined Hochschild and cyclic homology of $X$, respectively; that is, there are canonical isomorphisms $$HH_\bullet\big(\Perf(X)\big) \to HH_\bullet(X), \qquad HC_\bullet\big(\Perf(X)\big) \to HC_\bullet(X).$$

Choices:
B. They are canonically isomorphic to the geometrically defined Hochschild and cyclic homology of $X$ relative to the base field; that is, there are canonical isomorphisms $$HH_\bullet\big(\Perf(X)/\mathbb{K}\big) \to HH_\bullet(X/\mathbb{K}), \qquad HC_\bullet\big(\Perf(X)/\mathbb{K}\big) \to HC_\bullet(X/\mathbb{K}).$$
C. At least the Hochschild homology agrees with the geometrically defined Hochschild homology of $X$; namely, there is a canonical isomorphism $$HH_\bullet\big(\Perf(X)\big) \to HH_\bullet(X).$$
D. They are canonically isomorphic after passing to a flat or semi-free resolution of the structure sheaf and of the dg category; in particular, one has canonical isomorphisms only up to such resolutions, rather than directly for $\Perf(X)$ and $X$ themselves.
E. They are canonically isomorphic to the geometrically defined Hochschild and cyclic homology of $X$ for affine quasi-compact separated schemes, and more generally after imposing a flatness hypothesis on $\mathcal{O}_X$ over $\mathbb{K}$; namely, under those assumptions one has $$HH_\bullet\big(\Perf(X)\big) \to HH_\bullet(X), \qquad HC_\bullet\big(\Perf(X)\big) \to HC_\bullet(X).$$

The stem does not explicitly state the theorem’s conclusion or name the geometrically defined invariants as the answer. It only asks which invariants match the dg-category homologies, so there is no direct answer leakage.

This is essentially a theorem-recall question with mild reformulation. The correct option restates the standard comparison result, though the alternatives force attention to scope and hypotheses.

Some reasoning is required to reject subtle variants: relative vs. absolute homology, whether both HH and HC are included, and whether extra flatness/affineness assumptions are needed. Still, it mainly tests precise recall rather than deeper derivation.

The distractors are plausible and mathematically meaningful: confusing absolute with relative invariants, selecting a weaker true statement, adding unnecessary resolution caveats, or imposing extra hypotheses. These reflect common failure modes.

A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it is still fairly close to direct theorem recall rather than a genuinely generative reasoning task.


Real algebraic geometry can be thought of as a generalization of manifolds, where the continuous functions are replaced by polynomials with real coefficients. Application to physics also leads to a necessary generalization to associative algebraic geometry and a generalization of continuous Riemannian metrics, see the book of O.A. Laudal, \cite{Laudal21}. One of the main problems with this, is that Riemannian metrics are defined over the reals, and the polynomial algebra over the reals, $\mathbb R[x_1,\dots,x_n],$ contains more simple modules than $\mathbb R^n.$ Because the algebraic properties governing the simple modules are better controlled by an algebra over an algebraically closed field, the main result in this text is the construction of a $\mathbb C$-algebra $A_{\mathbb R}$ such that $\simp(A_{\mathbb R})\cong\mathbb R^n.$ Thus the points in $\mathbb R^n$ is in one-to-one correspondence with the simple $A_{\mathbb R}$-modules for which $\aspec(A_{\mathbb R})$ is a fine moduli, see the book \cite{S23} or the preprint \cite{S241}. For any real manifold $M$ we define an associative variety $(\mathcal M,\mathcal O^A_L)$ over $\mathbb C$ such that the points in $M$ is in bijective correspondence with the closed points in $\mathcal M,$ and such that the charts $U$ in $\mathcal M$ corresponds to $\mathcal O^A_L(U)=A_{\mathbb R}.$

In short terms, if we want to generalize the study of manifolds by using algebraic scheme-theory, we need schemes of associative $\mathbb C$-algebras for which the set of points is in bijective correspondence with $M.$

Because of the discovery of localization in associative rings, \cite{S252}, we include the general definition of associative schemes. Their purpose is to serve as moduli of algebraic objects which can be put in one-to-one correspondence with modules over associative schemes: That is, we need associative moduli to classify associative objects. This is most easily seen by the following.
Let $k$ be a field and consider the polynomial algebra in $n\geq 1$ variables $$A=k[x_1,\dots,x_n]=k[\underline x].$$ For a point $P=(p_1,\dots,p_n)\in k^n,$ we define the simple $A$-modules $$M(P)=A/(x_1-p_1,\dots,x_n-p_n).$$

\begin{abstract}
In the preprint \cite{S252} we proved that there exists a localizing ring $A_M$ for $A$ an associative ring with unit, and $M=\oplus_{i=1}^rM_i$ a direct sum of $r\geq 1$ simple right $A$-modules. For a homomorphism of associative rings $A\rightarrow B$ we define the contraction of a simple $B$-module to $A.$
Then we define the set of aprime right $A$-modules $\aspec A$ to be the set of simple $A$-modules together with contractions of such. When $A$ is commutative, $\aspec A=\spec A,$ and we define a topology on $\aspec A$ such that when $A$ is commutative, this is the Zariski topology. In the preprint \cite{S251}, we proved that when we have a topology and a localizing subcategory, there exists a sheaf of associative rings $\mathcal O_X$ on $\aspec A,$ agreeing with the usual sheaf of rings on $\spec A.$ In this text, we write out this construction, and we see that we can restrict the sheaf and topology to any subset $V\subseteq\aspec A.$ In particular, this
proves that we can use complex varieties in real algebraic geometry, by restricting in accordance with $\mathbb R\subseteq\mathbb C.$ Thus the theory of schemes over algebraically closed fields and its associative generalization can be applied to real (algebraic) geometry.
\end{abstract}

In short terms, if we want to generalize the study of manifolds by using algebraic scheme-theory, we need schemes of associative $\mathbb C$-algebras for which the set of points is in bijective correspondence with $M.$

\begin{proof} From \cite{S23} we know that $$\ext^1_A(M_1,M_2)\simeq\der_k(A,\hmm_k(M_1,M_2))/\inner$$ where $\hmm_k(M_1,M_2)$ is an $A-A$ bimodule by $a\phi(m)=\phi(am), \phi a(m)=\phi(m)a.$

When $P\neq Q$ we can consider $M_{\underline 0}, M_P$ with $p_1\neq 0.$
For $$\delta\in\der_k(A,\hmm_k(M_{\underline 0},M_P)$$ we have $$\begin{aligned}\delta(x_ix_1)&=\delta(x_1x_i)\\
&\Updownarrow\\
\delta(x_i)x_1+x_i\delta(x_1)&=\delta(x_1)x_i+x_1\delta(x_i)\\
&\Updownarrow\\
\delta(x_i)p_1&=\delta(x_1)p_i\\
&\Updownarrow\\
\delta(x_i)&=\delta(x_i)\frac{p_i}{p_1}.
\end{aligned}$$
This is also true for the inner derivations, proving that $\dim_k\inner=1$ so that $\ext^1_A(M_P,M_Q)=0$ when $P\neq Q.$
\end{proof}

\begin{proof} We have a homomorphism $f:A\rightarrow A/\mathfrak m$ such for every $s\in A\setminus\mathfrak m,$ $f(s)$ is a unit. By the universal property of localization, there exists a unique homomorphism $\phi:A_\mathfrak m\rightarrow A/\mathfrak m$ such that $\phi(\frac{a}{s})=\iota(a)\iota(s)^{-1}.$ This homomorphism is clearly surjective, and its kernel is $\mathfrak m A_\mathfrak m$ giving the wanted isomorphism.
\end{proof}

\begin{proof} We send the prime ideal $\mathfrak p\subset A$ to $A_\mathfrak p/\mathfrak p A_\mathfrak p$ which is $A$-prime by definition. On the other hand, let
$M$ be an aprime $A$-module, defined by $\iota_M: A\rightarrow B$ such that $M$ is a simple $B$-module. Let $\mathfrak m\subset B$ be the maximal ideal defining $M$ as a simple $B$-module. Then $\iota_M^{-1}(\mathfrak m)$ is a prime ideal in $A.$ That these two operations are inverses to each other follows from the fact that for a maximal ideal in a ring $B$ we have $B/\mathfrak m\simeq B_{\mathfrak m}/\mathfrak m B_\mathfrak m$ as proven i Lemma \ref{anotherloclemma}.
\end{proof}

\begin{definition} Define the subset of $k$-points in $\mathbb X$ by $$\tilde{\mathbb X}(k)=\{x\in X\subseteq\mathbb X|x\text{ is simple}\}\subseteq\mathbb X.$$ Then the induced associative subscheme $\mathbb X(k)$ is called the associative subscheme of $k$-points.
\end{definition}

\begin{proof} Because $k\subseteq\Bbbk$ is a sub-algebra, it follows that if $\phi\otimes\id$ is an isomorphism, then $\dim_k V_1=\dim_k V_2,$ and that choosing corresponding bases, $$0\neq\det(\phi\otimes\id)=\det\phi.$$
\end{proof}

\begin{proof} From \cite{S23} we know that $$\ext^1_A(M_1,M_2)\simeq\der_k(A,\hmm_k(M_1,M_2))/\inner$$ where $\hmm_k(M_1,M_2)$ is an $A-A$ bimodule by $a\phi(m)=\phi(am), \phi a(m)=\phi(m)a.$

When $P=Q$ we can assume $M_1=M_2=k[\underline x]/(\underline x).$ Then any inner derivation is on the form $\ad_\phi$ for which $\ad_\phi(x_i)=[\phi,x_i]=0.$ Thus the inner derivations is of dimension zero and $\ext^1_A(M_1,M_2)\simeq\der_k(A,k)$ where
$\operatorname d_i(x_i)=1,\ i=1,\dots,n$ gives a basis for the derivations.

When $P\neq Q$ we can consider $M_{\underline 0}, M_P$ with $p_1\neq 0.$
For $$\delta\in\der_k(A,\hmm_k(M_{\underline 0},M_P)$$ we have $$\begin{aligned}\delta(x_ix_1)&=\delta(x_1x_i)\\
&\Updownarrow\\
\delta(x_i)x_1+x_i\delta(x_1)&=\delta(x_1)x_i+x_1\delta(x_i)\\
&\Updownarrow\\
\delta(x_i)p_1&=\delta(x_1)p_i\\
&\Updownarrow\\
\delta(x_i)&=\delta(x_i)\frac{p_i}{p_1}.
\end{aligned}$$
This is also true for the inner derivations, proving that $\dim_k\inner=1$ so that $\ext^1_A(M_P,M_Q)=0$ when $P\neq Q.$
\end{proof}

It follows from the lemma that not all finite dimensional simple modules over a noncommutative $k$-algebra can be classified by a finitely generated commutative algebra. See \cite{S23} for a lot of examples.

From \cite{S23}: \(\ext^1_A(M_1,M_2)\simeq \der_k(A,\hmm_k(M_1,M_2))/\inner\), with \(\hmm_k(M_1,M_2)\) an \(A\)-\(A\) bimodule via \(a\phi(m)=\phi(am)\), \(\phi a(m)=\phi(m)a\).

- Case \(P=Q\): assume \(M_1=M_2=k[\underline x]/(\underline x)\). Any inner derivation is \(\ad_\phi\) and \(\ad_\phi(x_i)=[\phi,x_i]=0\), so \(\dim_k\inner=0\). Hence \(\ext^1_A(M_1,M_2)\simeq \der_k(A,k)\), and the derivations \(\operatorname d_i\) with \(\operatorname d_i(x_i)=1\) for \(i=1,\dots,n\) give a basis, so the dimension is \(n\).

- Case \(P\neq Q\): consider \(M_{\underline 0}, M_P\) with \(p_1\neq 0\). For \(\delta\in\der_k(A,\hmm_k(M_{\underline 0},M_P))\), comparing \(\delta(x_ix_1)=\delta(x_1x_i)\) yields
\[\delta(x_i)x_1+x_i\delta(x_1)=\delta(x_1)x_i+x_1\delta(x_i)\Rightarrow \delta(x_i)p_1=\delta(x_1)p_i\Rightarrow \delta(x_i)=\delta(x_1)\frac{p_i}{p_1}.\]
This relation also holds for inner derivations, and it is concluded that \(\dim_k\inner=1\), so \(\ext^1_A(M_P,M_Q)=0\) when \(P\neq Q\).

For two points $P,Q\in k^n$ we have that $$\dim_k\ext^1_A(M_P,M_Q)=\begin{cases}n,\ P=Q\\0,\ P\neq Q.\end{cases}$$,

Question:
Let $k$ be a field, let $A=k[x_1,\dots,x_n]$ with $n\ge 1$, and for each point $P=(p_1,\dots,p_n)\in k^n$ define the simple $A$-module
$$M_P:=A/(x_1-p_1,\dots,x_n-p_n).$$
Which statement correctly describes $\dim_k\operatorname{Ext}^1_A(M_P,M_Q)$ for two points $P,Q\in k^n$?

$$\dim_k\operatorname{Ext}^1_A(M_P,M_Q)=\begin{cases}n,& P=Q,\\0,& P\ne Q.\end{cases}$$

Choices:
B. $$\dim_k\operatorname{Ext}^1_A(M_P,M_Q)=\begin{cases}n,& P=Q,\\1,& P\ne Q.\end{cases}$$
C. $$\dim_k\operatorname{Ext}^1_A(M_P,M_Q)=0\quad\text{for all }P\ne Q.$$
D. $$\dim_k\operatorname{Ext}^1_A(M_P,M_Q)=\begin{cases}n-1,& P=Q,\\0,& P\ne Q.\end{cases}$$
E. $$\dim_k\operatorname{Ext}^1_A(M_P,M_Q)=\begin{cases}n,& P=Q,\\0,& P-Q\in (k^	imes)^n,\end{cases}$$

The stem defines the modules and asks for the Ext^1-dimension formula, but it does not reveal or strongly hint at the diagonal/off-diagonal values.

This is not a mere restatement in the stem; the respondent must choose among several competing formulas for both the P=Q and P≠Q cases.

To identify the correct choice, one must reason about Ext^1 between simples over a polynomial ring, including the vanishing off the diagonal and the n-dimensional tangent-space contribution on the diagonal.

The distractors are meaningfully different and plausibly reflect common errors: nonzero off-diagonal Ext, forgetting the diagonal case, miscounting the diagonal dimension as n-1, or imposing an irrelevant coordinatewise condition.

A strong MCQ: no answer leakage, non-tautological, and it tests genuine algebraic reasoning with mostly high-quality distractors.