L-systems and the Lov\'asz number

Given integers n > k > 0, and a set of integers L 0, k-1, an L-system is a family of sets F nk such that |F F'| L for distinct F, F' F. L-systems correspond to independent sets in a certain generalized Johnson graph G (n, k, L), so that the maximum size of an L-system is equivalent to finding the independence number of the graph G (n, k, L). The Lov\'asz number (G) is a semidefinite programming approximation of the independence number of a graph G. In this paper, we determine the order of magnitude of (G (n, k, L) ) of any generalized Johnson graph with k and L fixed and n. As an application of this theorem, we give an explicit construction of a graph G on n vertices with large gap between the Lov\'asz number and the Shannon capacity c (G). Specifically, we prove that for any > 0, for infinitely many n there is a generalized Johnson graph G on n vertices which has ratio (G) /c (G) = (n^1-), which greatly improves on the best known explicit construction.