Set families with a forbidden induced subposet

TL;DR

This paper determines the asymptotic maximum size of subset families avoiding a specific induced subposet, extending classical combinatorial results like Sperner's theorem to more complex poset structures.

Contribution

It generalizes Bukh's result to posets with a tree-shaped Hasse diagram of height k, providing a broad extension of known extremal set family bounds.

Findings

Maximum size asymptotic to (k-1) times the middle binomial coefficient

Extends Sperner's theorem to new classes of induced subposets

Provides a unified framework for forbidden induced subposet problems

Abstract

For each poset $H$ whose Hasse diagram is a tree of height $k$, we show that the largest size of a family $\cF$ of subsets of $[n]=\{1,..., n\}$ not containing $H$ as an induced subposet is asymptotic to $(k-1){n\choose \fl{n/2}}$. This extends the result of Bukh \cite{bukh}, which in turn generalizes several known results including Sperner's theorem.