Set families with a forbidden subposet

TL;DR

This paper determines the maximum size of subset families avoiding a specific tree-structured poset, extending classical combinatorial results like Sperner's theorem to a broader class of forbidden configurations.

Contribution

It provides an asymptotic characterization of the largest family avoiding a tree-shaped poset, generalizing known extremal set theory results.

Findings

Largest family size asymptotically determined for tree-shaped posets

Generalizes Sperner's theorem to new poset configurations

Establishes bounds for forbidden subposet problems

Abstract

We asymptotically determine the size of the largest family F of subsets of {1,...,n} not containing a given poset P if the Hasse diagram of P is a tree. This is a qualitative generalization of several known results including Sperner's theorem.