On a Diagonal Conjecture for Classical Ramsey Numbers

TL;DR

This paper investigates a conjecture about the behavior of classical Ramsey numbers, explores its implications, and connects it to limits involving the growth of these numbers, with potential consequences for understanding graph capacities.

Contribution

It provides evidence and implications of the Diagonal Conjecture for classical Ramsey numbers and links it to the limits of their exponential growth rates.

Findings

If the Diagonal Conjecture holds and a certain limit is finite, then all related limits are finite.

The work connects Ramsey number growth to Shannon capacity of graph complements.

It discusses conditions under which the limits of Ramsey numbers' roots are finite.

Abstract

Let $R(k_1, \cdots, k_r)$ denote the classical $r$-color Ramsey number for integers $k_i \ge 2$. The Diagonal Conjecture (DC) for classical Ramsey numbers poses that if $k_1, \cdots, k_r$ are integers no smaller than 3 and $k_{r-1} \leq k_r$, then $R(k_1, \cdots, k_{r-2}, k_{r-1}-1, k_r +1) \leq R(k_1, \cdots, k_r)$. We obtain some implications of this conjecture, present evidence for its validity, and discuss related problems. Let $R_r(k)$ stand for the $r$-color Ramsey number $R(k, \cdots, k)$. It is known that $\lim_{r \rightarrow \infty} R_r(3)^{1/r}$ exists, either finite or infinite, the latter conjectured by Erd\H{o}s. This limit is related to the Shannon capacity of complements of $K_3$-free graphs. We prove that if DC holds, and $\lim_{r \rightarrow \infty} R_r(3)^{1/r}$ is finite, then $\lim_{r \rightarrow \infty} R_r(k)^{1/r}$ is finite for every integer $k \geq 3$.