On families of subsets with a forbidden subposet

TL;DR

This paper introduces a novel approach combining extremal graph theory and probability to determine the maximum size of families of subsets avoiding certain posets, expanding known results for specific classes like cycles and trees.

Contribution

It applies new methods to asymptotically determine the maximum size of H-free families for classes of posets including cycles, two-end-forks, and up-down trees.

Findings

Asymptotic formulas for maximum sizes of H-free families for specific posets

Identification of new classes of posets with known extremal sizes

Application of graph theory and probability methods to poset problems

Abstract

Let $\F\subset 2^{[n]}$ be a family of subsets of $\{1,2,..., n\}$. For any poset $H$, we say $\F$ is $H$-free if $\F$ does not contain any subposet isomorphic to $H$. Katona and others have investigated the behavior of $\La(n,H)$, which denotes the maximum size of $H$-free families $\F\subset 2^{[n]}$. Here we use a new approach, which is to apply methods from extremal graph theory and probability theory to identify new classes of posets $H$, for which $\La(n,H)$ can be determined asymptotically as $n\to\infty$ for various posets $H$, including two-end-forks, up-down trees, and cycles $C_{4k}$ on two levels.