2026/07/16 by Nils Bengone, Amine Brouk, Max Grinsztajn +4
#math.CO
We present an extension of the template-based approach for Schur numbers developed by Rowley. This new form of template construction, which we call shifted S-templates, was discovered during a conversation with ChatGPT 5.5 Pro, then refined, verified, and extended to multiple shifted S-templates. These new templates generalize the first ones by giving more flexibility in the coloring. Using this added flexibility with the new way to color the special label cells of the template, we exhibit a template which yields the recurrence S(k+2) ≥ 10S(k)+2, improving on the classical Abbott-Hanson recurrence S(k+2) ≥ 9S(k)+4 for the same step. Combined with the known bounds S(6) ≥ 536 and S(11) ≥ 203 828, this implies S(8) ≥ 5 362 and S(13) ≥ 2 038 282, improving the previously listed lower bounds 5 286 and 2 011 290.