An upper bound for the three-colour Ramsey number of the ten-cycle: R(C10,C10,C10) <= 26

Abstract

We prove that R(C10, C10, C10) <= 26. The best upper bound recorded in theDynamic Survey of Small Ramsey Numbers (DS1, revision 18, April 2026) wasR3(C10) <= 3015, a specialization of a 1975 theorem of Erdos and Graham; thesurvey names R3(C10) as the first open case of Dzido's conjectureR3(C2m) = 4m. The interval was [20, 3015] and is now [20, 26], against aconjectured value of 20. We also record the weaker bound R3(C10) <= 27, whichfollows from counting alone. The proof combines a counting argument, the extremal C10-free graphs of order26 determined by Afzaly and McKay (seven graphs, all block graphs with blocksK9, K9, K9, K2 and independence number at most 4), and an exhaustive SATcomputation: in any 3-colouring of K26 with no monochromatic C10 some colourclass has exactly 109 edges and is one of the seven; the remaining 216 edgesare then shown, exhaustively, not to split into two C10-free classes of 108edges each. The definitive campaign runs, per instance, three steps on one retained cubefile: the cube generator to completion, an explicit refutation of the formulaoutside the cubes, and the refutation of every cube — 786,095 cubes in total,zero timeouts, zero models. Two earlier independently generated campaigns(817,028 and 1,016,065 cubes) agree. The retained artifacts — cube files,per-cube solver outputs, coverage outputs — are deposited with SHA-256 hashes,and a CHECK.sh script re-verifies the deposit from a fresh extraction. The deposit contains the note, the SAT encodings, the seven extremal graphsbyte-identical to Afzaly and McKay's file, the patch to SAT Modulo Symmetriesneeded to rebuild the solver, all drivers, the raw logs of all campaignsincluding two aborted attempts, the independent verification scripts, and aworking notebook. The README states explicitly which checks were performed andwhich were not. No DRAT/LRAT proof certificate was produced; this is declaredin the note as the limit of what the deposit establishes.