It’s an extension of Schur’ Theorem. I need to proof that there exist $N$ such that for any coloring of first $N$ natural numbers in three colors there will be three one colored numbers $x, y, z$ such that $x + y = 2z$.
Edit. I came up with this equation when I was trying to prof that W(3, 3) is finite (i.e. there exist n such that for every 3-coloring of first n natural numbers one can always find an ariphmetic progression of three numbers x < z < y (equivalent to y - z = z - x <=> x + y = 2z)).