On a Diagonal Conjecture for classical Ramsey numbers (original) (raw)

2019, Discrete Applied Mathematics

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.