S(6) ≤ 1801 if R₄(3) ≤ 61: a centred Schur bound and the structure at the frontier
A Lean 4 formalization, with Mathlib, of an upper bound for the Schur numbers in terms of the triangle Ramsey numbers, and of the structure of a colouring at the limit of that bound. If every colouring with k colours of the pairs of r points has a monochromatic triangle (R_k(3) ≤ r), then the interval [1, 2((k + 1)⌊(r − 1)/2⌋ + 1)] is not covered by k + 1 sumfree sets, so S(k + 1) ≤ 2(k + 1)⌊(r − 1)/2⌋ + 1. With R_2(3) ≤ 6 this gives S(3) ≤ 13, which is exact. Under the hypothesis R_4(3) ≤ 61, a computer-assisted claim of M. Tatarevic that this project does not prove, the pigeonhole step…
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