The method of double chains for largest families with excluded subposets

TL;DR

This paper introduces a new method for determining the maximum size of subset families avoiding a given poset, providing exact results for many cases and bounds for all, using intersection counting with double chains.

Contribution

It develops a novel double chain intersection counting method to exactly determine or bound the largest families avoiding specific posets.

Findings

Exact values of La(n,P) for infinitely many P constructed from seven base posets.

An upper bound for La(n,P) based on |P| and the longest chain in P.

Introduction of the double chains intersection counting technique.

Abstract

For a given finite poset $P$, $La(n,P)$ denotes the largest size of a family $\mathcal{F}$ of subsets of $[n]$ not containing $P$ as a weak subposet. We exactly determine $La(n,P)$ for infinitely many $P$ posets. These posets are built from seven base posets using two operations. For arbitrary posets, an upper bound is given for $La(n,P)$ depending on $|P|$ and the size of the longest chain in $P$. To prove these theorems we introduce a new method, counting the intersections of $\mathcal{F}$ with double chains, rather than chains.