 
Summary: ON MONOCHROMATIC SUBSETS OF A RECTANGULAR GRID
MARIA AXENOVICH AND JACOB MANSKE
Abstract. For n N, let [n] denote the integer set {0, 1, . . . , n  1}. For any subset V Z2
, let
Hom(V ) = {cV + b : c N, b Z2
}. For k N, let Rk(V ) denote the least integer N0 such that
for any N N0 and for any kcoloring of [N]2
, there is a monochromatic subset U Hom(V ).
The argument of Gallai ensures that Rk(V ) is finite. We investigate bounds on Rk(V ) when V is a
three or fourpoint configuration in general position. In particular, we prove that R2(S) V W (8),
where V W is the classical van der Waerden number for arithmetic progressions and S is a square
S = {(0, 0), (0, 1), (1, 0), (1, 1)}.
1. Introduction
Let, for a positive integer n, [n] = {0, 1, . . . , n  1}. The classical Theorem of van der Waerden [16]
claims that for any n, k N, there is N0 N such that for all N N0 and any kcoloring : [N] [k],
there is a monochromatic arithmetic progression of length n (nAP). Define V W(k, n) to be the least
such integer guaranteed by van der Waerden's Theorem. The number V W(n) = V W(2, n) is usually
referred to as the classical van der Waerden number. The best known bounds are
(n  1)2n1
V W(n) 222222n+9
