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An improved lower bound on the Shannon capacities of complements of odd cycles

Daniel G. Zhu

TL;DR

This work addresses the Shannon capacity of the complements of odd cycles $\\bar{C}_{2n+1}$ by developing a combinatorial box framework (good boxes, skeletons, expansions) to construct large independent sets in $\\bar{C}_{2n+1}^{\\boxtimes r}$. It proves that with $r_n$ equal to the number of partitions of $2(n-1)$ into powers of $2$, one has $\\alpha(\\bar{C}_{2n+1}^{\\boxtimes r_n}) \\ge 2^{r_n}+1$, which yields the lower bound $\\Theta(\\bar{C}_{2n+1}) \\ge (2^{r_n}+1)^{1/r_n} = 2+\\Omega(2^{-r_n}/r_n)$; asymptotically, $\\log r_n \\sim (\\log n)^2/(2\\log 2)$. This improves on the doubly exponential Bohman–Holzman bound and connects to Day and Johnson’s work on multicolor graph Ramsey numbers, via a relation between homomorphic Ramsey numbers and Shannon capacity. The paper also clarifies the scope and limitations of good-box constructions and provides an appendix with proofs of auxiliary results. The results advance understanding of nontrivial lower bounds for the Shannon capacity of nontrivial graph complements and suggest new directions via Ramsey-theoretic connections.

Abstract

Improving a 2003 result of Bohman and Holzman, we show that for $n \geq 1$, the Shannon capacity of the complement of the $2n+1$-cycle is at least $(2^{r_n} + 1)^{1/r_n} = 2 + Ω(2^{-r_n}/r_n)$, where $r_n = \exp(O((\log n)^2))$ is the number of partitions of $2(n-1)$ into powers of $2$. We also discuss a connection between this result and work by Day and Johnson in the context of graph Ramsey numbers.

An improved lower bound on the Shannon capacities of complements of odd cycles

TL;DR

This work addresses the Shannon capacity of the complements of odd cycles by developing a combinatorial box framework (good boxes, skeletons, expansions) to construct large independent sets in . It proves that with equal to the number of partitions of into powers of , one has , which yields the lower bound ; asymptotically, . This improves on the doubly exponential Bohman–Holzman bound and connects to Day and Johnson’s work on multicolor graph Ramsey numbers, via a relation between homomorphic Ramsey numbers and Shannon capacity. The paper also clarifies the scope and limitations of good-box constructions and provides an appendix with proofs of auxiliary results. The results advance understanding of nontrivial lower bounds for the Shannon capacity of nontrivial graph complements and suggest new directions via Ramsey-theoretic connections.

Abstract

Improving a 2003 result of Bohman and Holzman, we show that for , the Shannon capacity of the complement of the -cycle is at least , where is the number of partitions of into powers of . We also discuss a connection between this result and work by Day and Johnson in the context of graph Ramsey numbers.
Paper Structure (3 sections, 14 equations, 1 figure)

This paper contains 3 sections, 14 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:main']}
  3. Further Remarks Concerning Good Boxes

Figures (1)

  • Figure 1: \ref{['ex1']} The $(3;4,6)$-expansion $\psi$ defined in the proof of \ref{['prop:ex']}. Filled circles represent the range of $\psi$ inside $B(4,6)$, and an edge is drawn between $\psi(x,i)$ and $\psi(y,j)$ for all $x,y\in B(3)$ and $i,j\in [2]$ with $x \sim y$ and $(x,i) \neq (y,j)$. \ref{['ex2']} The $(3;4,4)$-expansion defined in \ref{['rmk:bh']}, depicted similarly to \ref{['ex1']}.

Theorems & Definitions (7)

  • proof
  • proof
  • proof
  • proof
  • proof : Proof of \ref{['thm:main']}
  • proof
  • proof