The Three-Color Generalized Schur Number of x+(k−1)y=z: A Proof for All k from a Critical Set of 33 Polynomials
Abstract
For an integer k≥ 3, let Tk(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 Tk(3)=k3+2k2−2 for all k≥ 3. In the notation sa(3) of the first author, with a=k−1, this says that sa(3)=a3+5a2+7a+1 for all a≥ 2, which confirms the conjecture proposed in 2020. Note that every solution to x+(k−1)y=z is also a solution to x1+…+xk=xk+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,k3+2k2−3].
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BibTeX
@misc{khamseh2026threecolor,
title = {{The Three-Color Generalized Schur Number of \(x+(k-1)y=z\): A Proof for All \(k\) from a Critical Set of 33 Polynomials}},
author = {Amir Khamseh and Mohammad Hassan Safareh},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01714},
eprint = {2611.01714},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01714},
}
RIS
TY - UNPB
TI - The Three-Color Generalized Schur Number of x+(k−1)y=z: A Proof for All k from a Critical Set of 33 Polynomials
AU - Amir Khamseh
AU - Mohammad Hassan Safareh
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01714
UR - https://arxiv.org/abs/2611.01714
AB - For an integer k≥ 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≥ 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≥ 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+…+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].
ER -