On crown-free families of subsets

TL;DR

This paper investigates the maximum size of subset families avoiding certain cyclic poset structures, extending known results to odd cycles of length at least 14, and confirms asymptotic bounds for these crown-free families.

Contribution

It proves that the maximum size of crown-free families matches the asymptotic bound for all odd cycles of length at least 14, expanding previous results limited to even cycles.

Findings

Confirmed asymptotic maximum size for odd cycles of length ≥14

Extended previous bounds from even to certain odd cycles

Established that crown-free families are asymptotically half of the power set size

Abstract

The crown $\Oh_{2t}$ is a height-2 poset whose Hasse diagram is a cycle of length $2t$. A family $\F$ of subsets of $[n]:=\{1,2..., n\}$ is {\em $\Oh_{2t}$-free} if $\Oh_{2t}$ is not a weak subposet of $(\F,\subseteq)$. Let $\La(n,\Oh_{2t})$ be the largest size of $\Oh_{2t}$-free families of subsets of $[n]$. De Bonis-Katona-Swanepoel proved $\La(n,\Oh_{4})= {n\choose \lfloor \frac{n}{2} \rfloor} + {n\choose \lceil \frac{n}{2} \rceil}$. Griggs and Lu proved that $\La(n,\Oh_{2t})=(1+o(1))\nchn$ for all even $t\ge 4$. In this paper, we prove $\La(n,\Oh_{2t})=(1+o(1))\nchn$ for all odd $t\geq 7$.