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 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 vertical bar P vertical bar and the size of the longest chain in P. To prove these theorems we introduce a new method, counting the intersections of F with double chains, rather than chains.