The Abbott-Hanson Recurrence for Schur Numbers
Prior state unknown→proved
improves the classical recurrence; the exact Schur numbers beyond S(5) remain unknown
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combinatorics / Ramsey theory
The Abbott-Hanson Recurrence for Schur Numbers
The classical Abbott-Hanson recurrence gives $S(k+2) \ge 9S(k)+4$ for Schur numbers, and had stood as the basis for the best asymptotic lower bounds. Shifted $S$-templates, a more flexible form of Rowley's template construction, yield $S(k+2) \ge 10S(k)+2$ and hence improved lower bounds.
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The classical Abbott-Hanson recurrence gives $S(k+2) \ge 9S(k)+4$ for Schur numbers, and had stood as the basis for the best asymptotic lower bounds. Shifted $S$-templates, a more flexible form of Rowley's template construction, yield $S(k+2) \ge 10S(k)+2$ and hence improved lower bounds.
improves the classical recurrence; the exact Schur numbers beyond S(5) remain unknown
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