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arXiv · 2410.13758

On monochromatic solutions to linear equations over the integers

Dingding Dong·Nitya Mani·Huy Tuan Pham·Jonathan Tidor
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Abstract

We study the number of monochromatic solutions to linear equations in a $2$-coloring of $\{1,\ldots,n\}$. We show that any nontrivial linear equation has a constant fraction of solutions that are monochromatic in any $2$-coloring of $\{1,\ldots,n\}$. We further study commonness of four-term equations and disprove a conjecture of Costello and Elvin by showing that, unlike over $\mathbb{F}_p$, the four-term equation $x_1 + 2x_2 - x_3 - 2x_4 = 0$ is uncommon over $\{1,\ldots,n\}$.

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Dingding Dong, Nitya Mani, Huy Tuan Pham, Jonathan Tidor. 2024-10-27. On monochromatic solutions to linear equations over the integers. https://arxiv.org/abs/2410.13758

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