Intersecting $P$-free families

TL;DR

This paper investigates the maximum size of intersecting families avoiding a given poset, providing exact results for specific cases like the butterfly poset and general bounds for others, using novel combinatorial methods.

Contribution

It introduces a new generalization of the partition method to determine the size of largest intersecting P-free families and classifies equality cases for the butterfly poset.

Findings

Exact size of largest intersecting B-free family determined.

General bounds established for intersecting P-free families.

New proof for maximum size of intersecting k-Sperner families.

Abstract

We study the problem of determining the size of the largest intersecting $P$-free family for a given partially ordered set (poset) $P$. In particular, we find the exact size of the largest intersecting $B$-free family where $B$ is the butterfly poset and classify the cases of equality. The proof uses a new generalization of the partition method of Griggs, Li and Lu. We also prove generalizations of two well-known inequalities of Bollob\'{a}s and Greene, Katona and Kleitman in this case. Furthermore, we obtain a general bound on the size of the largest intersecting $P$-free family, which is sharp for an infinite class of posets originally considered by Burcsi and Nagy, when $n$ is odd. Finally, we give a new proof of the bound on the maximum size of an intersecting $k$-Sperner family and determine the cases of equality.