Bounds on Shannon Capacity and Ramsey Numbers from Product of Graphs

TL;DR

This paper explores the relationship between Shannon capacity and Ramsey numbers, providing new lower bounds and showing limitations of finite graph powers in achieving maximum capacity.

Contribution

It improves existing constructions for multicolor Ramsey numbers and demonstrates that the supremum of Shannon capacity cannot be attained by finite graph powers.

Findings

New lower bounds for multicolor Ramsey numbers

Supremum of Shannon capacity not achievable by finite graph powers

Generalization to graphs with any bounded independence number

Abstract

In this note we study Shannon capacity of channels in the context of classical Ramsey numbers. We overview some of the results on capacity of noisy channels modelled by graphs, and how some constructions may contribute to our knowledge of this capacity. We present an improvement to the constructions by Abbott and Song and thus establish new lower bounds for a special type of multicolor Ramsey numbers. We prove that our construction implies that the supremum of the Shannon capacity over all graphs with independence number 2 cannot be achieved by any finite graph power. This can be generalized to graphs with any bounded independence number.