Lexicographic Ramsey Theory

Given positive integers d and n, there is an integer N such that for every injective map ffrom { 1 ..... N} a into R there is a subset A = A1 x A2 x ... × Ad of { 1 ..... N} a such that (1) each Aj 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 texicographic 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. © 1993 Academic Press, Inc.