On a Diagonal Conjecture for classical Ramsey numbers

摘要

Let R(k(1), ..., k(r)) denote the classical r-color Ramsey number for integers k(i) >= 2. The Diagonal Conjecture (DC) for classical Ramsey numbers poses that if k(1), ..., k(r) are integers no smaller than 3 and k(r-1 )<= k(r), then R(k(1), ..., k(r-2), k(r-1) - 1, k(r) + 1) <= R(k(1), ..., k(r)). We obtain some implications of this conjecture, present evidence for its validity, and discuss related problems.