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

We prove that R(C10, C10, C10) <= 26. The best upper bound recorded in the
Dynamic Survey of Small Ramsey Numbers (DS1, revision 18, April 2026) was
R3(C10) <= 3015, a specialization of a 1975 theorem of Erdos and Graham; the
survey names R3(C10) as the first open case of Dzido's conjecture
R3(C2m) = 4m. The interval was [20, 3015] and is now [20, 26], against a
conjectured value of 20. We also record the weaker bound R3(C10) <= 27, which
follows from counting alone.

The proof combines a counting argument, the extremal C10-free graphs of order
26 determined by Afzaly and McKay (seven graphs, all block graphs with blocks
K9, K9, K9, K2 and independence number at most 4), and an exhaustive SAT
computation: in any 3-colouring of K26 with no monochromatic C10 some colour
class has exactly 109 edges and is one of the seven; the remaining 216 edges
are then shown, exhaustively, not to split into two C10-free classes of 108
edges each.

The definitive campaign runs, per instance, three steps on one retained cube
file: the cube generator to completion, an explicit refutation of the formula
outside 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 graphs
byte-identical to Afzaly and McKay's file, the patch to SAT Modulo Symmetries
needed to rebuild the solver, all drivers, the raw logs of all campaigns
including two aborted attempts, the independent verification scripts, and a
working notebook. The README states explicitly which checks were performed and
which were not. No DRAT/LRAT proof certificate was produced; this is declared
in the note as the limit of what the deposit establishes.