Generalized Schur Numbers for $x_1 + x_2 + c = 3x_3$

Abstract

Let $r(c)$ be the least positive integer $n$ such that every two coloring of the integers $1,\ldots,n$ contains a monochromatic solution to $x_1 + x_2 + c = 3x_3$. Verifying a conjecture of Martinelli and Schaal, we prove that $$ r(c) = \left\lceil{2\lceil{2+c\over3}\rceil + c\over3}\right\rceil, $$ for all $c \ge 13$, and $$ r(c) = \left\lceil{3\lceil{3-c\over2}\rceil - c\over2}\right\rceil, $$ for all $c \le -4$.