On crown-free families of subsets

The crown O 2 t 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 O 2 t -free if O 2 t is not a weak subposet of ( F , ⊆ ) . Let La ( n , O 2 t ) be the largest size of O 2 t -free families of subsets of [ n ] . De Bonis–Katona–Swanepoel proved La ( n , O 4 ) = ( n ⌊ n 2 ⌋ ) + ( n ⌈ n 2 ⌉ ) . Griggs and Lu proved that La ( n , O 2 t ) = ( 1 + o ( 1 ) ) ( n ⌊ n 2 ⌋ ) for all even t ≥ 4 . In this paper, we prove La ( n , O 2 t ) = ( 1 + o ( 1 ) ) ( n ⌊ n 2 ⌋ ) for all odd t ≥ 7 .