Pairwise Intersections and Forbidden Configurations

Abstract

Let $f_m(a,b,c,d)$ denote the maximum size of a family $\mathcal{F}$ of subsets of an $m$-element set for which there is no pair of subsets $A,B \in \mathcal{F}$ with $|A \cap B| \geq a$, $|\bar{A} \cap B| \geq b$, $|A \cap \bar{B}| \geq c$, and $|\bar{A} \cap \bar{B}| \geq d$. By symmetry we can assume $a \geq d$ and $b \geq c$. We show that $f_m(a,b,c,d)$ is $\Theta (m^{a+b-1})$ if either $b > c$ or $a,b \geq 1$. We also show that $f_m(0,b,b,0)$ is $\Theta (m^b)$ and $f_m(a,0,0,d)$ is $\Theta (m^a)$. This can be viewed as a result concerning forbidden configurations and is further evidence for a conjecture of Anstee and Sali. Our key tool is a strong stability version of the Complete Intersection Theorem of Ahlswede and Khachatrian, which is of independent interest.

Keywords

• forbidden configurations
• extremal set theory
• intersecting set systems
• uniform set systems
• (0 1)-matrices
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]
• [math.math-co] mathematics [math]/combinatorics [math.co]