$L$-systems and the Lovász number

Given integers $n>k>0$, and a set of integers $L\subset [0,k-1]$, an \emph{$L$-system} is a family of sets $\mathcal{F}\subset \left(\genfrac{}{}{0px}{}{[n]}{k}\right)$ such that $\mathrm{\mid }F\cap F^{\mathrm{\prime }}\mathrm{\mid }\in L$ for distinct $F,F^{\mathrm{\prime }}\in \mathcal{F}$. $L$-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 \emph{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 $n\to \mathrm{\infty }$. 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 $\vartheta (G)\mathrm{/}c(G)=\mathrm{\Omega }(n^{1-\epsilon })$, which improves on all known constructions. The graph $G$ $\mathit{\text{a}}\mathit{\text{fortiori}}$ also has ratio $\vartheta (G)\mathrm{/}\alpha (G)=\mathrm{\Omega }(n^{1-\epsilon })$, which greatly improves on the best known explicit construction.