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

Improving a 2003 result of Bohman and Holzman, we show that for $n\ge 1$, the Shannon capacity of the complement of the $2n+1$-cycle is at least $(2^{r_n}+1{)}^{1\mathrm{/}{r}_n}=2+\mathrm{\Omega }(2^{-r_n}\mathrm{/}{r}_n)$, 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.