Schur-like numbers and a lemma of Shearer

TL;DR

This paper establishes upper bounds on the size of number sets colored with n colors that avoid monochromatic solutions to specific additive equations, improving previous bounds and exploring related equations.

Contribution

It provides new asymptotic bounds for colorings avoiding solutions to certain equations, extending and strengthening prior results in combinatorial number theory.

Findings

Bound N=O((n!)^{1/2}) for no monochromatic solutions to x1+x2+x3=y1+y2.

Stronger bound N=O(((n-k)!)^{1/2}) for equations with more variables.

Remarks on other equations and their solution bounds.

Abstract

Suppose that each number $1,2,...,N$ has one of n colours assigned. We show that if there are no monochromatic solutions to the equation $x_1+x_2+x_3=y_1+y_2$, then $N=O((n!)^{1/2})$, improving upon a result of Cwalina and Schoen. Further, a stronger bound of $N=O(((n-k)!)^{1/2})$, where $k\gg\frac{\log n}{\log\log n}$ is shown for colourings avoiding solutions to the equation $x_1+x_2+...+x_{12}=y_1+y_2+...+y_9$. Finally, some remarks on other equations are presented.