A note on strongly and totally chain intersecting families

TL;DR

This paper extends previous work on chain intersecting families by determining the maximum size of strongly and totally chain intersecting families for large sets, providing new exact bounds.

Contribution

It precisely determines the maximum size of strongly $(p,q)$-chain intersecting families for large $n$, and of totally $(2,2)$-chain intersecting families, advancing the understanding of these combinatorial structures.

Findings

Largest cardinality of strongly $(p,q)$-chain intersecting families for large $n$

Maximum size of totally $(2,2)$-chain intersecting families

Extension of previous partial results to exact bounds

Abstract

Bern\'ath and Gerbner in 2007 introduced $(p,q)$-chain intersecting families of subsets of an $n$-element underlying set. Those have the property that for any $p$-chain $A_1\subsetneq A_2\subsetneq \dots \subsetneq A_p$ and $q$-chain $B_1\subsetneq B_2\subsetneq \dots \subsetneq B_q$, we have $A_p\cap B_q\neq \emptyset$. Bern\'ath and Gerbner determined the largest cardinality of such families. They also introduced strongly $(p,q)$-chain intersecting families, where $A_p\cap B_1\neq \emptyset$ and totally $(p,q)$-chain intersecting families, where $A_1\cap B_1\neq \emptyset$. They obtained some partial results on the maximum cardinality of such families. We extend those results by determining the largest cardinality of strongly $(p,q)$-chain intersecting families if $n$ is sufficiently large, and by determining the largest cardinality of totally $(2,2)$-chain intersecting families.