L-Systems and the Lovász Number

Abstract

Given integers \(n> k > 0\), and a set of integers \(L \subset [0, k-1]\), an L-system is a family of sets \(\mathcal {F}\subset \left( {\begin{array}{c}[n]\\ k\end{array}}\right) \) such that \(|F \cap F'| \in L\) for distinct \(F, F'\in \mathcal {F}\). 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 Lovász number \(\vartheta (G)\) is a semidefinite programming approximation of the independence number \(\alpha \) of a graph G. In this paper, we determine the leading order term of \(\vartheta (G(n, k, L))\) of any generalized Johnson graph with k and L fixed and \(n\rightarrow \infty \). As an application of this theorem, we give an explicit construction of a graph G on n vertices with a large gap between the Lovász number and the Shannon capacity c(G). Specifically, we prove that for any \(\epsilon > 0\), for infinitely many n there is a generalized Johnson graph G on n vertices which has ratio \(\vartheta (G)/c(G) = \Omega (n^{1-\epsilon })\), which improves on all known constructions. The graph G a fortiori also has ratio \(\vartheta (G)/\alpha (G) = \Omega (n^{1-\epsilon })\), which greatly improves on the best known explicit construction.