Given integers $n > k > 0$, and a set of integers $L \subset [0, k-1]$, an $L$-system is a family of sets $\mathcal{F} \subset \binom{[n]}{k}$ 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 Lov\'asz number $\vartheta(G)$ is a semidefinite programming approximation of the independence number of a graph $G$. In this paper, we determine the order of magnitude 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 large gap between the Lov\'asz 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 greatly improves on the best known explicit construction.

翻译：给定整数 $n > k > 0$ 以及整数集合 $L \subset [0, k-1]$，一个 $L$-系统是指满足如下条件的集族 $\mathcal{F} \subset \binom{[n]}{k}$：对任意不同的 $F, F'\in \mathcal{F}$，有 $|F \cap F'| \in L$。$L$-系统对应于某个广义Johnson图 $G(n, k, L)$ 中的独立集，因此最大 $L$-系统的大小等价于求图 $G(n, k, L)$ 的独立数。Lovász数 $\vartheta(G)$ 是图 $G$ 独立数的一个半定规划近似。本文中，我们确定了当 $k$ 和 $L$ 固定且 $n\rightarrow \infty$ 时，任意广义Johnson图 $G(n, k, L)$ 的 $\vartheta(G(n, k, L))$ 的数量级。作为该定理的一个应用，我们显式构造了一个 $n$ 个顶点的图 $G$，其Lovász数与Shannon容量 $c(G)$ 之间存在巨大间隔。具体地，我们证明了对任意 $\epsilon > 0$，存在无穷多个 $n$ 以及一个 $n$ 个顶点的广义Johnson图 $G$，使得比值 $\vartheta(G)/c(G) = \Omega(n^{1-\epsilon})$，这大幅改进了已知的最佳显式构造结果。