On monochromatic solutions to linear equations over the integers

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 $\{1,\ldots,n\}$.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Proposition 3.1
• proof
• proof : Proof of \ref{['thm:lower-sys']}
• Proposition 4.1
• proof
• Corollary 4.2