The Three-Color Generalized Schur Number of \(x+(k-1)y=z\): A Proof for All \(k\) from a Critical Set of 33 Polynomials
For an integer \(k\ge 3\), let \(T_k(3)\) be the least positive integer \(N\) such that, for any \(3\)-coloring of \([1,N]\), there exists a monochromatic solution to \(x+(k-1)y=z\). In this paper, we prove that \(T_k(3)=k^3+2k^2-2\) for all \(k\ge 3\). In the notation \(s_a(3)\) of the first author, with \(a=k-1\), this says that \(s_a(3)=a^3+5a^2+7a+1\) for all \(a\ge 2\), which confirms the conjecture proposed in 2020. Note that every solution to \(x+(k-1)y=z\) is also a solution to \(x_1+\dots+x_k=x_{k+1}\). So the theorem also gives a new proof of the upper bound in the three-color theorem of Boza, Marín, Revuelta and Sanz, using only this restricted equation. The lower bound is an explicit recursive coloring. For the upper bound, we fix one set \(\Xmin\) of \(33\) polynomials in \(k\), together with the \(144\) polynomial identities \(z=x+(k-1)y\) that hold among them, and we show that no \(3\)-coloring of \(\Xmin\) avoids a monochromatic identity. Since every relation is an identity in \(k\), a single argument covers all \(k\). The argument is a case analysis with \(62\) cases and depth \(8\), and it uses only two elementary rules. It is written out in full in an appendix and has been checked by an independent program. The set \(\Xmin\) is critical, that is, removing any one of its elements makes it \(3\)-colorable. It is also a critical subset of minimum size of the \(51\)-element set from which it was extracted. Finally, we describe the rigid structure shared by all extremal colorings of \([1,k^3+2k^2-3]\).