Given integers $n > k > 0$, and a set of integers $L \subset [0, k-1]$, an \emph{$L$-system} is a family of sets $\mathcal{F} \subset \binom{[n]}{k}$ 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 \emph{Lov\'asz 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 Lov\'asz number and the Shannon capacity $c(G)$. Specifically, we prove that for any $\epsilon > 0$, for sufficiently large $n$ 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 $G$ \textit{a fortiori} also has ratio $\vartheta(G)/\alpha(G) = \Omega(n^{1-\epsilon})$, 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 \emph{$L$-系统} 是集合族 $\mathcal{F} \subset \binom{[n]}{k}$，使得对任意不同的 $F, F'\in \mathcal{F}$ 有 $|F \cap F'| \in L$。$L$-系统对应于特定广义约翰逊图 $G(n, k, L)$ 中的独立集，因此 $L$-系统的最大规模等价于求图 $G(n, k, L)$ 的独立数。\emph{洛瓦兹数} $\vartheta(G)$ 是图 $G$ 独立数 $\alpha$ 的半定规划近似。本文中，我们确定了对于任意固定 $k$ 和 $L$ 且 $n\rightarrow \infty$ 的广义约翰逊图 $G(n, k, L)$ 中 $\vartheta(G(n, k, L))$ 的首项阶数。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 $G$，其洛瓦兹数与香农容量 $c(G)$ 之间存在大间隙。具体地，我们证明了对任意 $\epsilon > 0$，当 $n$ 充分大时，存在一个 $n$ 个顶点的广义约翰逊图 $G$ 使得比值 $\vartheta(G)/c(G) = \Omega(n^{1-\epsilon})$，这一结果改进了所有已知构造。该图 $G$ \textit{进而}也满足比值 $\vartheta(G)/\alpha(G) = \Omega(n^{1-\epsilon})$，这大幅改进了已知的最佳显式构造。