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$L$-systems and the Lovász number

William Linz

TL;DR

This work studies L-systems as independent sets in generalized Johnson graphs $G(n,k,L)$ and uses the Lovász number $\vartheta(G)$ as a tractable proxy for the independence number and Shannon capacity. By leveraging the Johnson scheme and Schrijver's LP reformulation, the authors establish that for fixed $k$ and $L$, $\vartheta(G(n,k,L))$ grows as $\Theta(n^{|L|})$ and provide explicit constants tied to the run structure of $L$. They also connect these results to bounds on Shannon capacity via the Haemers/minrank framework, and construct explicit families (notably $G_q(n,k)$ with $k=q^{2}-1$) that exhibit large gaps between $\vartheta(G)$ and $c(G)$, achieving $\vartheta(G)/c(G)=\Omega(N^{1-\varepsilon})$ for infinitely many $n$ (where $N$ is the number of vertices). The combination of association-scheme LP techniques and algebraic-minrank bounds yields both tight asymptotics for $\vartheta$ in the generalized Johnson family and strong, explicit gaps to Shannon capacity, advancing understanding of when Lovász-type bounds closely approximate or vastly exceed true information-theoretic limits.

Abstract

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{Lovász number} $\vartheta(G)$ is a semidefinite programming approximation of the independence number $α$ 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 $ε> 0$, for infinitely many $n$ there is a generalized Johnson graph $G$ on $n$ vertices which has ratio $\vartheta(G)/c(G) = Ω(n^{1-ε})$, which improves on all known constructions. The graph $G$ \textit{a fortiori} also has ratio $\vartheta(G)/α(G) = Ω(n^{1-ε})$, which greatly improves on the best known explicit construction.

$L$-systems and the Lovász number

TL;DR

This work studies L-systems as independent sets in generalized Johnson graphs and uses the Lovász number as a tractable proxy for the independence number and Shannon capacity. By leveraging the Johnson scheme and Schrijver's LP reformulation, the authors establish that for fixed and , grows as and provide explicit constants tied to the run structure of . They also connect these results to bounds on Shannon capacity via the Haemers/minrank framework, and construct explicit families (notably with ) that exhibit large gaps between and , achieving for infinitely many (where is the number of vertices). The combination of association-scheme LP techniques and algebraic-minrank bounds yields both tight asymptotics for in the generalized Johnson family and strong, explicit gaps to Shannon capacity, advancing understanding of when Lovász-type bounds closely approximate or vastly exceed true information-theoretic limits.

Abstract

Given integers , and a set of integers , an \emph{-system} is a family of sets such that for distinct . -systems correspond to independent sets in a certain generalized Johnson graph , so that the maximum size of an -system is equivalent to finding the independence number of the graph . The \emph{Lovász number} is a semidefinite programming approximation of the independence number of a graph . In this paper, we determine the leading order term of of any generalized Johnson graph with and fixed and . As an application of this theorem, we give an explicit construction of a graph on vertices with a large gap between the Lovász number and the Shannon capacity . Specifically, we prove that for any , for infinitely many there is a generalized Johnson graph on vertices which has ratio , which improves on all known constructions. The graph \textit{a fortiori} also has ratio , which greatly improves on the best known explicit construction.
Paper Structure (6 sections, 16 theorems, 96 equations)

This paper contains 6 sections, 16 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. The Lovász number of generalized Johnson graphs
  3. Background on association schemes
  4. Analyzing the linear program for the Lovász number of generalized Johnson graphs
  5. The Haemers bound and the minrank parameter
  6. Explicit constructions of large gaps between the Shannon capacity and the Lovász number

Key Result

Theorem 1.2

Let $G = G(n, k, L)$ be a generalized Johnson graph for some integers $n > k > 0$ and a set $L = \{\ell_1, \ell_2, \ell_3, \ldots, \ell_s\} \subset [0, k-1]$ with $\ell_1 < \ell_2 < \ldots < \ell_s$. Suppose $L$ contains $b$ full runs of consecutive integers. Then, there is a constant $c$ depending and a constant $C$ depending on $k$ and $L$, but not on $n$ such that where $m_i$ is the length of

Theorems & Definitions (36)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3: Deza--Erdős--Frankl DEZ
  • Theorem 1.4: Ray-Chaudhuri--Wilson RCW
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1: Association Scheme
  • Definition 2.2: Johnson scheme
  • Definition 2.3: Lovász number
  • Theorem 2.4: Schrijver Sch1979
  • ...and 26 more