The four-colour triangle Ramsey number
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R(3,3,3,3) is the least number of vertices that forces a monochromatic triangle in every four-colouring of a complete graph’s edges. Determining it requires matching upper and lower bounds.
Why it matters and what counts as progress
Why it matters: This small finite target tests how structural graph arguments and exhaustive computation can establish unavoidable patterns.
What counts as progress: Improve either endpoint of the interval, determine the exact value, or provide fully checked certificates for the remaining computational exclusions.
How the problem developed
- 2004 · Research result [1]
Susan Fettes, Richard Kramer and Stanisław Radziszowski Proved the upper bound R(3,3,3,3)≤62.
- 24 Apr 2026 · Research result [1]
Stanisław Radziszowski The Small Ramsey Numbers survey retained the interval 51≤R(3,3,3,3)≤62.
- 27 Oct 2026 · AI contribution [1]
Milos Tatarevic with GPT and Claude models Reported an upper bound of 61. Lean checks the reductions, while 56,723 SAT exclusions remain external solver obligations.
AI contribution
An upper bound of 61 for the Ramsey number R(3,3,3,3)
Tatarevic reports that every four-colouring of the edges of K₆₁ contains a monochromatic triangle, lowering the upper bound for R(3,3,3,3) from 62 to 61. The argument combines incidence counting, finite neighbourhood classifications and 56,723 SAT exclusions.
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