On monochromatic solutions to linear equations over the integers

We study the number of monochromatic solutions to linear equations in a $2$-coloring of $\{1,\dots ,n\}$. We show that any nontrivial linear equation has a constant fraction of solutions that are monochromatic in any $2$-coloring of $\{1,\dots ,n\}$. 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 $\{1,\dots ,n\}$.