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
This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.
Publication Date
7-22-2019
Document Type
Article
Department, Program, or Center
Computer Science (GCCIS)
Recommended Citation
Liang, Meilian; Radziszowski, Stanislaw; and Xu, Xiaodong, "On a Diagonal Conjecturefor Classical Ramsey Numbers" (2019). Discrete Applied Mathematics, 267 (), 195--200. Accessed from
https://repository.rit.edu/article/1937
Campus
RIT – Main Campus
Comments
DOI: 10.1016/j.dam.2019.07.006