Lexicographic Ramsey theory

Given positive integers d and n , there is an integer N such that for every injective map f from {1, …, N } d into R there is a subset A = A 1 × A 2 × … × A d of {1, …, N } d such that (1) each A j has n elements, (2) the restriction of f to A is monotone in each coordinate, (3) there is an ordering of the coordinates such that f on A is lexicographic with respect to that ordering. Because injection f is otherwise arbitrary, the direction of monotonicity for each coordinate (increasing or decreasing) and the coordinate ordering for item (3) cannot be prespecified. Results on necessary sizes of N are included.