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

TL;DR

This paper improves the lower bounds on the Shannon capacities of complements of odd cycles, revealing new asymptotic behavior and connecting to graph Ramsey numbers, advancing understanding in information theory and combinatorics.

Contribution

The paper establishes a significantly improved lower bound on the Shannon capacity of complements of odd cycles, extending previous results from 2003.

Findings

Lower bound of $(2^{r_n} + 1)^{1/r_n}$ for Shannon capacity

Asymptotic relation: $2 + oldsymbol{ ext{Omega}}(2^{-r_n}/r_n)$

Connection to graph Ramsey numbers

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 $(2^{r_n} + 1)^{1/r_n} = 2 + \Omega(2^{-r_n}/r_n)$, 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.