Year: 2026
Type: preprint
Abstract: We prove that the generalized threshold Ramsey number R(5,10;5,8) equals 21. This number corresponds to the set Ramsey number r_5(10,8) introduced by Harborth. The proof is computer-assisted. We first enumerate all (4,6;5,8)-good and (5,10;4,4)-good graphs. Then we apply the lucky-vertex inequality derived from the McKay--Radziszowski edge identity to filter candidate pairs for a gluing search, and encode the remaining cases as SAT instances. The search yields exactly 17 non-isomorphic (5,10;5,8;20)-good graphs. Finally, we exclude all one-vertex extensions of these 17 graphs, which proves that no (5,10;5,8;21)-good graph exists.Additionally, we update the tabulation of small set Ramsey numbers r_5(m,n) for 1 <= m,n <= 10.
Source: arXiv (Cornell University)
Authors: Guo, Wancai, Wan, Min
Institution: Shihezi University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223240853