On a Diagonal Conjecture for classical Ramsey numbers

Abstract

Let R(k1,…,kr) denote the classical r-color Ramsey number for integers ki≥2. The Diagonal Conjecture (DC) for classical Ramsey numbers poses that if k1,…,kr are integers no smaller than 3 and kr−1≤kr, then R(k1,…,kr−2,kr−1−1,kr+1)≤R(k1,…,kr). We obtain some implications of this conjecture, present evidence for its validity, and discuss related problems.Let Rr(k) stand for the r-color Ramsey number R(k,…,k). It is known that limr→∞Rr(3)1∕r exists, either finite or infinite, the latter conjectured by Erdős. This limit is related to the Shannon capacity of complements of K3-free graphs. We prove that if DC holds, and limr→∞Rr(3)1∕r is finite, then limr→∞Rr(k)1∕r is finite for every integer k≥3.