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 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 $G$ \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$n > k > 0$及整数集合$L \subset [0, k-1]$，\emph{L-系统}是指集合族$\mathcal{F} \subset \binom{[n]}{k}$，满足对任意不同的$F, F'\in \mathcal{F}$有$|F \cap F'| \in L$。L-系统对应于特定广义Johnson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{Lovász数}$\vartheta(G)$是图$G$独立数$\alpha$的半定规划逼近。本文确定了当$k$和$L$固定且$n\rightarrow \infty$时，任意广义Johnson图$\vartheta(G(n, k, L))$的首项阶。作为该定理的应用，我们显式构造了具有$n$个顶点的图$G$，其Lovász数与香农容量$c(G)$之间存在巨大间隙。具体而言，我们证明对任意$\epsilon > 0$，存在无穷多个$n$使得$n$个顶点上的广义Johnson图$G$满足比值$\vartheta(G)/c(G) = \Omega(n^{1-\epsilon})$，该结果改进了所有已知构造。该图\textit{a fortiori}亦满足比值$\vartheta(G)/\alpha(G) = \Omega(n^{1-\epsilon})$，这显著改进了当前最优的显式构造结果。