关于3元一致U(s,q)集族的最大基数

On the maximum size of 3-uniform U(s,q)families

假设n、k、s和q为正整数,n>q≥k,sk>q,s≥2.给定一个集族F?(k[n]),如果对于任意F1,…,Fs∈F,都有|F1∪…∪Fs|≤q,则称F是一个U(s,q)集族.这个概念由Frankl和Kupavskii(2021)引入.它是两类常见集族的推广:(1)t-交族;(2)最多有s个成员互不相交的集族.Frankl和Kupavskii(2021)提出如下问题:决定U(s,q)集族的最大基数.本文充分研究k=3的情形,并且在s≥s0(t)时,确定U(s,2s+t)集族的最大基数.特别地,本文证明Frankl和Kupavskii(2021)提出的一个关于3元一致U(s,q)集族的最大基数的猜想.

Let n,k,s,q be integers,n>q≥k,sk>q and s≥2.A family F?(k[n])is U(s,q)if|F1∪…∪Fs|≤q for any F1,…,Fs∈F.This notion was introduced recently by Frankl and Kupavskii(2021)and it generalizes the property of a family being t-intersecting,or having no more than s pairwise disjoint members.Frankl and Kupavskii(2021)formulated a problem of finding the maximum size of U(s,q)families.In this paper,we study the k=3 case extensively,and determine the maximum size of U(s,2s+t)families for all s≥s0(t)and all meaningful n and q.In particular,we prove a conjecture proposed by Frankl and Kupavskii(2021)on the maximum size of3-uniform U(s,q)families.