The method of double chains for largest families with excluded subposets

For a given finite poset $P$, $La(n,P)$ denotes the largest size of a family $\mathcal{F}$ of subsets of $[n]$ not containing $P$ as a weak subposet. We exactly determine $La(n,P)$ for infinitely many $P$ posets. These posets are built from seven base posets using two operations. For arbitrary posets, an upper bound is given for $La(n,P)$ depending on $|P|$ and the size of the longest chain in $P$. To prove these theorems we introduce a new method, counting the intersections of $\mathcal{F}$ with double chains, rather than chains.