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 subposet P. This problem has been studied intensively in recent years, and it is conjectured that $\pi(P):= \lim_{n\rightarrow\infty} La(n,P)/{n choose n/2}$ exists for general posets P, and, moreover, it is an integer. For $k\ge2$ 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 \pi(\D_k) for infinitely many values $k$. A stubborn open problem is to show that \pi(\D_2)=2; here we make progress by proving \pi(\D_2)\le 2 3/11 (if it exists).Comment: 16 page