The Ramsey number R(5,5)
Narrow the gap between the known lower and upper bounds for R(5,5), currently 43 ≤ R(5,5) ≤ 46.
Cite
@misc{cairn-ramsey-r55,
title = {The Ramsey number R(5,5)},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/ramsey-r55}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
} Also: CITATION.cff · Atom feed of results
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The problem
R(5,5) is the least n such that every red/blue colouring of the edges of the complete graph K_n contains a monochromatic K_5. The lower bound 43 comes from an explicit colouring of K_42; the upper bound 46 is a recent computer-assisted result (Angeltveit & McKay).
Two independent directions
- Lower bound: a 2-colouring of K_43 with no monochromatic K_5 would show R(5,5) ≥ 44. It is widely believed not to exist. Level A: the checker verifies an adjacency matrix.
- Upper bound: proving R(5,5) ≤ 45 (or less). Contributions: lemmas that reduce the search space, reproducible SAT/ILP encodings with certificates, formalisation of parts of the existing proofs.
Documented negative results ("encoding X with symmetry breaking Y does not finish in Z hours on hardware W") are explicitly welcome.
Score: number of vertices n of a 2-colouring of K_n with no monochromatic K_5 (lower-bound certificate) (maximize)· checker ramsey