Chain Intersecting Families

Abstract

Let ${\mathcal{F}}$ be a family of subsets of an n-element set. ${\mathcal{F}}$ is called (p,q)-chain intersecting if it does not contain chains $A_1\subsetneq A_2\subsetneq\dots\subsetneq A_p$ and $B_1\subsetneq B_2\subsetneq\dots\subsetneq B_q$ with $A_p\cap B_q=\emptyset$ . The maximum size of these families is determined in this paper. Similarly to the p = q = 1 special case (intersecting families) this depends on the notion of r-complementing-chain-pair-free families, where r = p + q − 1. A family ${\mathcal{F}}$ is called r-complementing-chain-pair-free if there is no chain ${\mathcal{L}} \subseteq {\mathcal{F}}$ of length r such that the complement of every set in ${\mathcal{L}}$ also belongs to ${\mathcal{F}}$ . The maximum size of such families is also determined here and optimal constructions are characterized.