Schur-like numbers and a lemma of Shearer

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.