A Computer-Assisted Proof that R(5,10;5,8)=21 for the Generalized Threshold Ramsey Number
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.
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