Two-color Rado Number for $x+y+c=4z$

Abstract

For positive integers $c\ge 0$ and $k\ge 1$, let $n=R(c,k)$ be the least integer, provided it exists, such that every $2$-coloring of the set $[1,n]=\{1,\dots ,n\}$ admits a monochromatic solution to the equation $x+y+c=4z$ with $x,y,z\in [1,n]$. In this paper, the precise value of $R(c,4)$ is shown to be $\lceil 3c+2/8\rceil$ for all even $c\ge 34$.