Intersection-union families

Abstract

Let $2^{\left[n\right]}$ denote the power set of the n-set $\left[n\right]=\{1,2,\dots ,n\}$. For positive integers $n,p,q$, $n\ge p+q$ let $m(n,p,q)$ denote the maximum of $\left|F\right|$ for a family $F\subset 2^{\left[n\right]}$ satisfying $|F\cap G|\ge p$ and $|F\cup G|\le n-q$ for all $F,G\in F$. The exact value of $m(n,p,q)$ has been known for half a century in the case $p=1$ or $q=1$. Bang, Sharp and Winkler determined it in the case $n-p-q\le 3$. The aim of the present paper is to establish the exact value of $m(n,p,q)$ for $n\ge {(n-p-q+1)}^3$ and also for $n-p-q=4$.