On a Diagonal Conjecture for Classical Ramsey Numbers

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 ${\lim }_{r\to \mathrm{\infty }}{R}_r(3{)}^{1\mathrm{/}r}$ 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 ${\lim }_{r\to \mathrm{\infty }}{R}_r(3{)}^{1\mathrm{/}r}$ is finite, then $\lim_{r \rightarrow \infty} R_r(k)^{1/r}$isfiniteforeveryinteger$k \geq 3$.