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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.\n  Let $R_r(k)$ stand for the $r$-color Ramsey number $R(k, \\cdots, k)$. 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