Diamond-free families

Given a finite poset P, we consider the largest size La ( n , P ) of a family of subsets of [ n ] : = { 1 , … , n } that contains no (weak) subposet P. This problem has been studied intensively in recent years, and it is conjectured that π ( P ) : = lim n → ∞ La ( n , P ) / ( n ⌊ n 2 ⌋ ) exists for general posets P, and, moreover, it is an integer. For k ⩾ 2 let D k denote the k-diamond poset { A < B 1 , … , B k < C } . We study the average number of times a random full chain meets a P-free family, called the Lubell function, and use it for P = D k to determine π ( D k ) for infinitely many values k. A stubborn open problem is to show that π ( D 2 ) = 2 ; here we make progress by proving π ( D 2 ) ⩽ 2 3 11 (if it exists).