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 Erd˝os. 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.

Creative Commons License

Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.

Publication Date

7-22-2019

Comments

DOI: 10.1016/j.dam.2019.07.006

Document Type

Article

Department, Program, or Center

Computer Science (GCCIS)

Campus

RIT – Main Campus

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