AI 中文总结
研究1974年Erdős和Kleitman提出的求极大\(k -\)元相交族最小规模问题，改进了之前Janzer的上界，将首项系数对\(k\)的依赖从指数降为线性，与已知下界相差因子\(4\)。
AI 中文摘要
一个\([n] := \{1,2,\ldots, n\}\)的子集族\(\mathcal{F}\)，如果\(\mathcal{F}\)中任意至多\(k\)个成员的集合都有非空交集，且向\(\mathcal{F}\)添加任何其他集合都会破坏此性质，那么\(\mathcal{F}\)被称为极大\(k -\)元相交族。1974年Erdős和Kleitman提出的一个老问题是求极大\(k -\)元相交族的最小规模。\(k = 3\)时对于所有足够大的\(n\)已知结果，但\(k \geqslant 4\)时问题仍未解决。之前最著名的上界由Janzer给出，对于足够大且能被\(k - 1\)整除的\(n\)，其首项为\((k - 1)2^{k - 3}2^{n/(k - 1)}\)。本文将此界改进为\((4k - 10)2^{n/(k - 1)}\)，将首项系数对\(k\)的依赖从指数型降至线性型，且与已知下界相差一个\(4\)的因子。
英文摘要
A family $\mathcal{F}$ of subsets of $[n] := \{1,2,\ldots, n\}$ is called maximal $k$-wise intersecting if every collection of at most $k$ members of $\mathcal{F}$ has a non-empty intersection, and adding any other set to $\mathcal{F}$ breaks this property. An old question by Erdős and Kleitman from 1974 asks for the minimum size of a maximal $k$-wise intersecting family. The case $k = 3$ is known for all sufficiently large $n$, but the problem remains open for all $k \geqslant 4$. The previous best-known upper bound is by Janzer, which has a leading term $(k-1)2^{k-3}2^{n/(k-1)}$ for sufficiently large $n$ divisible by $k-1$. In this note, we improve this bound to $(4k-10)2^{n/(k-1)}$, which reduces the dependence on $k$ in the leading coefficient from exponential to linear and is within a factor of $4$ of the known lower bound.
Comments8 pages, 1 figure. Comments are welcome
