On a Diagonal Conjecture for Classical Ramsey Numbers [article]

Meilian Liang, Stanisław Radziszowski, Xiaodong Xu
2019 arXiv   pre-print
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. Let R_r(k) stand for the r-color Ramsey number R(k, ..., k). It is known that lim_r →∞ R_r(3)^1/r exists,
more » ... ither finite or infinite, the latter conjectured by Erdős. This limit is related to the Shannon capacity of complements of K_3-free graphs. We prove that if DC holds, and lim_r →∞ R_r(3)^1/r is finite, then lim_r →∞ R_r(k)^1/r is finite for every integer k ≥ 3.
arXiv:1810.11386v2 fatcat:tr352ks2i5bc3ddm3cd5t6adnq