🤖 AI Summary
This paper improves the Shannon capacity lower bound for complements of odd cycles, breaking the constant-factor precision barrier established by Bohman and Holzman (2003). Methodologically, it constructs the first explicit lower bound of the form $(2^{r_n}+1)^{1/r_n}$, where $r_n$ denotes the 2-adic partition number of $2(n-1)$, satisfying $r_n = exp(O((log n)^2))$. This yields an asymptotic lower bound of $2 + Omega(2^{-r_n}/r_n)$, substantially surpassing prior results. The approach integrates combinatorial graph theory, analysis of graph products, integer partitions, and asymptotic estimation, while uncovering a deep connection to Ramsey theory. Crucially, this is the first work to explicitly link the Shannon capacity lower bound to integer partition structure and achieve superexponential precision improvement—marking a qualitative advance in the quantitative understanding of this fundamental information-theoretic parameter.
📝 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.