What question did this study set out to answer?

Q: What question did this study set out to answer?

Establish an improved upper bound for the three-colour Ramsey number of the ten-cycle, R(C10, C10, C10).

August 20, 2026Open Access

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

View Full Paper

Ask AI

Bookmark

Authors

NGNicolás Federico Galindez

Discussion

Member takes

Implication

Exhaustive computational proof demonstrates that the three-colour Ramsey number for the ten-cycle is at most 26, narrowing the search interval from [20, 3015] toward Dzido's conjecture.

Key Points

Establish an improved upper bound for the three-colour Ramsey number of the ten-cycle, R(C10, C10, C10).
Combined theoretical edge-counting arguments with the seven extremal C10-free block graphs of order 26 classified by Afzaly and McKay.
Formulated an exhaustive Boolean satisfiability (SAT) framework using cube-and-conquer methods and SAT Modulo Symmetries to prove no valid 3-colouring of K26 avoids a monochromatic C10.
Executed a definitive campaign evaluating 786,095 cubes to completion alongside two independent verification campaigns (817,028 and 1,016,065 cubes) with public SHA-256 artifacts.
Proved that R(C10, C10, C10) <= 26, narrowing the previously open interval of [20, 3015] down to [20, 26].
Derived a weaker analytical upper bound of R3(C10) <= 27 directly from counting arguments alone.
Confirmed zero satisfying models across all 786,095 cubes in the exhaustive SAT refutation.

Cite This Study

Nicolás Federico Galindez (2026) studied this question.

synapsesocial.com/papers/6a86b5978a91293e6a1cd07f https://doi.org/10.5281/zenodo.21999999

View Full Paper

Ask AI

Bookmark

Implication

Exhaustive computational proof demonstrates that the three-colour Ramsey number for the ten-cycle is at most 26, narrowing the search interval from [20, 3015] toward Dzido's conjecture.

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

Discussion

Member takes

Implication

Key Points

Cite This Study

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

Discussion

Member takes

Implication

Key Points

Cite This Study

Also Consider

Also Consider