Intersecting non-uniform families containing subfamilies

TL;DR

This paper investigates the maximum size of intersecting non-uniform families of sets with specific size constraints, extending classical results and employing advanced combinatorial theorems.

Contribution

It determines the maximum size of intersecting families within certain non-uniform set families, introducing new bounds and methods in extremal combinatorics.

Findings

Maximum size of intersecting families in binom{[n]}{a}  binom{[2n]}{b} for n > b

Maximum size in complex unions of set families for large n

Results are close to optimal, using advanced combinatorial theorems

Abstract

A family of sets is said to be intersecting if every pair of sets in the family have non-empty intersection. In this paper, we initiate the study of intersecting non-uniform families of sets of one of two sizes containing given subfamilies. For a set $X$ and integer $r$, let $\binom{X}{r}$ denote the family $\{A \subseteq X: |X| = r\}$. Let $a$, $b$, and $n$ be positive integers such that $a < b$. We determine the maximum size of an intersecting family in $\binom{[n]}{a} \cup \binom{[2n]}{b}$ whenever $n > b$. For $n$ sufficiently large, we also determine the maximum size of an intersecting family in $\binom{[2n]}{a} \cup \binom{[n+1, 3n]}{a} \cup \binom{[n] \cup [2n + 1, 3n]}{a} \cup \binom{[3n]}{b}$ whenever $3n > 2b$ and $b > a + 2$. Our results are, in some sense, best possible. Our methods include the use of Katona's shadow intersection theorem and a recent diversity theorem of Kupavskii and~Zakharov.