  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 $k \geq 3$.
