  Let $\F\subset 2^{[n]}$ be a family of subsets of $\{1,2,..., n\}$. For any
poset $H$, we say $\F$ is $H$-free if $\F$ does not contain any subposet
isomorphic to $H$. Katona and others have investigated the behavior of
$\La(n,H)$, which denotes the maximum size of $H$-free families $\F\subset
2^{[n]}$. Here we use a new approach, which is to apply methods from extremal
graph theory and probability theory to identify new classes of posets $H$, for
which $\La(n,H)$ can be determined asymptotically as $n\to\infty$ for various
posets $H$, including two-end-forks, up-down trees, and cycles $C_{4k}$ on two
levels.
