On crown-free families of subsets
arXiv:1206.6258
Abstract
The crown $\Oh_{2t}$ is a height-2 poset whose Hasse diagram is a cycle of length . A family $\F$ of subsets of 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 . 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 . In this paper, we prove $\La(n,\Oh_{2t})=(1+o(1))\nchn$ for all odd .
17 pages