发表机构
Mathematical Institute, University of Oxford（牛津大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明3AP相交族的大小至多为$(\tfrac12-c)2^n$，首次非平凡逼近Simonovits-Sós猜想，并推广到有界共度的3-图情形。
AI 中文摘要
设$F$是$[n]$的子集族，若其中任意两个成员的相交部分包含一个非平凡的三项等差数列，则称$F$为3AP相交族。我们证明存在绝对常数$c>0$，使得任何这样的$F$的大小至多为$(\tfrac12 - c)2^n$。这是对Simonovits和Sós猜想（最大可能大小为$2^{n-3}$）的首个非平凡进展。更一般地，我们证明当$H$是$[n]$上具有有界共度的3-图时，同样的界对$H$-相交族也成立。一个团表明该结果是尖锐的，即有界共度假设不能被移除。
英文摘要
A family $F$ of subsets of $[n]$ is 3AP-intersecting if every two members have intersection containing a non-trivial three-term arithmetic progression. We prove that there is an absolute constant $c>0$ such that any such $F$ has size at most $(\tfrac12 - c)2^n$. This is the first non-trivial progress towards a conjecture of Simonovits and Sós that the maximum possible size is $2^{n-3}$. More generally, we show the same bound for $H$-intersecting families whenever $H$ is a $3$-graph on $[n]$ with bounded codegrees. A clique shows that this is sharp, in that the bounded codegree assumption cannot be removed.
Comments3 pages
