A direct approach to the Gallai-Schur numbers
Fred Rowley
TL;DR
Problem: determine exact $GS(r)$ and $WGS(r)$ and the structure of maximal Gallai-Schur partitions for both strong and weak variants. Method: develop two constructive liftings $^{2}\Theta$ and $^{5}\Theta$, analyze maximal partitions under a color-order and a modulo-$5$ residue structure, and derive closed-form expressions for $GS(r)$ and $WGS(r)$ with respect to the parity of $r$. Contributions: exact, parity-dependent formulas for $GS(r)$ and $WGS(r)$; a complete, unique-for-even-$r$ classification of maximal partitions; and a self-contained proof extending Budden's strong-case results to the weak case (where $WGS(r)=(9/5)\,GS(r)$ for $r>1$). Significance: deepens understanding of sum-free colorings on intervals and provides explicit bounds that are directly applicable to combinatorial number theory questions surrounding monochromatic and rainbow $a+b=c$ configurations.
Abstract
This paper characterises the structure of every maximal weak or strong Gallai-Schur partition. The results confirm the exact values of Gallai-Schur numbers provided by Budden (2020) in the strong case, and provide corresponding values for weak Gallai-Schur numbers. The proofs are elementary and standalone.