## On crown-free families of subsets

- Sept. 12, 2012
- 3:30 p.m.
- LeConte 312

## Abstract

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