On crown-free families of subsets

The crown O2t 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 O2t-free if O2t is not a weak subposet of (F,⊆). Let La(n,O2t) be the largest size of O2t-free families of subsets of [n]. De Bonis-Katona-Swanepoel proved La(n,O4)=(n⌊n2⌋)+(n⌈n2⌉). Griggs and Lu proved that La(n,O2t)=(1+o(1))(n⌊n2⌋) for all even t≥4. In this paper, we prove La(n,O2t)=(1+o(1))(n⌊n2⌋) for all odd t≥7.