On a Diagonal Conjecture for Classical Ramsey Numbers

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 $k_{r-1}\le k_r$, then $R(k_1,\cdots ,k_{r-2},k_{r-1}-1,k_r+1)\le R(k_1,\cdots ,k_r)$. 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 ${\lim }_{r\to \infty }{R}_r(3{)}^{1/r}$ exists, either finite or infinite, the latter conjectured by Erdős. This limit is related to the Shannon capacity of complements of $K_3$-free graphs. We prove that if DC holds, and ${\lim }_{r\to \infty }{R}_r(3{)}^{1/r}$ is finite, then ${\lim }_{r\to \infty }{R}_r(k{)}^{1/r}$ is finite for every integer $k\ge 3$.

10 pages