Poset-free Families and Lubell-boundedness

Given a finite poset $P$, we consider the largest size \lanp of a family \F of subsets of $[n]:=\{1,...,n\}$ that contains no subposet $P$. This continues the study of the asymptotic growth of \lanp; it has been conjectured that for all $P$, \pi(P):= \lim_{n\rightarrow\infty} \lanp/\nchn exists and equals a certain integer, $e(P)$. While this is known to be true for paths, and several more general families of posets, for the simple diamond poset \D_2, the existence of $\pi$ frustratingly remains open. Here we develop theory to show that $\pi(P)$ exists and equals the conjectured value $e(P)$ for many new posets $P$. We introduce a hierarchy of properties for posets, each of which implies $\pi=e$, and some implying more precise information about \lanp. The properties relate to the Lubell function of a family \F of subsets, which is the average number of times a random full chain meets \F. We present an array of examples and constructions that possess the properties