I want to create a bipartite graph where the first partite $U$ contains $L$ vertices with degree $k$ and the second partite $V$ contains $N$ other vertices with degree $a$. I need to find the minimum number of $N$ so that every pair of the $L$ vertices have a common neighbour.
A different formulation of my problem is the following:
Assume we have $L$ distinct numbers. I want to calculate the minimum number of sets of size $a$ that are needed so that all unordered pairs of numbers are contained in at least one of the sets. I understand that finding the sets is similar to the set cover problem: http://en.wikipedia.org/wiki/Set_cover_problem (right?), but is there an easier/faster way to calculate the minimum number without calculating the sets, given $L$ and $a$?