On crown-free families of subsets

The crown ${\text{Oh}}_{2t}$ is a height-2 poset whose Hasse diagram is a cycle of length $2t$. A family $\text{F}$ of subsets of $[n]:=\{1,\mathrm{2...},n\}$ is {\em ${\text{Oh}}_{2t}$-free} if ${\text{Oh}}_{2t}$ is not a weak subposet of $(\text{F},\subseteq )$. Let $\text{La}(n,{\text{Oh}}_{2t})$ be the largest size of ${\text{Oh}}_{2t}$-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 $\text{La}(n,{\text{Oh}}_{2t})=(1+o(1))\text{nchn}$ for all even $t\ge 4$. In this paper, we prove $\text{La}(n,{\text{Oh}}_{2t})=(1+o(1))\text{nchn}$ for all odd $t\ge 7$.