发表机构
School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对Erdős–Kleitman问题，构造了1≤k≤m−1时的无穷多新极值族，验证了Frankl–Kupavskii元猜想，并对相关结果进行了定量扩展。
AI 中文摘要
对于整数n≥s≥2，令e(n,s)为满足不存在s个两两不交成员的族F⊆2^{[n]}的最大规模，该问题现称为Erdős–Kleitman问题，与著名的Erdős匹配问题密切相关。Frankl与Kupavskii提出了一个元猜想，预测该最大值总是由加权族达到。固定m≥3，记n=ms+c（0≤c<s），并设ℓ=s−c。对于0≤k≤m，令a_k=ms−kc−1。对于A∈binom{[n]}{a_k}，定义H^k(m,s,ℓ;A):={F⊆[n]:k|F|+|F∩A|≥m(k+1)}，这定义了一类匹配数小于s的统一加权族。此前已知H^0、H^1和H^m在不同的c范围内是极值族。我们证明，对于1≤k≤m−1，所有族H^k在某些c范围内是唯一的极值族。更确切地说，我们证明对于每个m≥3和每个1≤k≤m−1，存在常数α=α(m,k)>0、β=β(m,k)>0和整数s₀=s₀(m,k)，使得对于所有整数s≥s₀及所有满足0≤c<s的整数c，当βs^{(k−1)/k}≤c≤αs^{k/(k+1)}时，e(n,s)的唯一极值族为A∈binom{[n]}{a_k}的族H^k(m,s,ℓ;A)。该结果确定了Erdős–Kleitman问题的无穷多新极值族，并在这些范围内验证了Frankl–Kupavskii元猜想，还对Kupavskii与Sokolov关于H^1极值性的结果提供了定量扩展。
英文摘要
For integers $n\ge s\ge2$, let $e(n,s)$ be the maximum size of a family $\mathcal F\subseteq2^{[n]}$ with no $s$ pairwise disjoint members. The problem of determining $e(n,s)$, now called the Erdős--Kleitman problem, is closely related to the well-known Erdős matching problem. Frankl and Kupavskii posed a meta-conjecture predicting that the maximum is always attained by a weighted family. Fix $m\ge3$, write $n=ms+c$ with $0\le c<s$, and set $\ell=s-c$. For $0\le k\le m$, let $a_k=ms-kc-1$. For $A\in\binom{[n]}{a_k}$, define \[ \mathcal H^k(m,s,\ell;A):= \{F\subseteq[n]: k|F|+|F\cap A|\ge m(k+1)\}. \] This defines a unified class of weighted families with matching number less than $s$. Among these families, $\mathcal H^0$, $\mathcal H^1$, and $\mathcal H^m$ were previously known to be extremal in different ranges of $c$. We show that for $1\le k\le m-1$, all families $\mathcal H^k$ are uniquely extremal in some ranges of $c$. More precisely, we prove that for every $m\ge3$ and every $1\le k\le m-1$, there exist constants $α=α(m,k)>0$, $β=β(m,k)>0$ and an integer $s_0=s_0(m,k)$ such that, for all integers $s\ge s_0$ and all integers $c$ with $0\le c<s$, the only extremal families for $e(n,s)$ are the families $\mathcal H^k(m,s,\ell;A)$ with $A\in\binom{[n]}{a_k}$ whenever $βs^{(k-1)/k}\le c\le αs^{k/(k+1)}$. In particular, this result determines an infinite number of new extremal families for the Erdős--Kleitman problem and verifies the Frankl--Kupavskii meta-conjecture in these ranges. This also provides a quantitative extension of the result of Kupavskii and Sokolov on the extremality of $\mathcal H^1$.
