Monochromatic Boxes in Colored Grids

A d-dimensional grid is a set of the form L 1⁄4 1⁄2a1 × · · · ×1⁄2ad , where 1⁄2t 1⁄4 f1; : : : ; tg and aj is a positive integer for j ∈ 1⁄2d . A d-dimensional box is a set of the form fx1; y1g× · · · ×fxd; ydg for some integers xj, yj with j ∈ 1⁄2d , where xj ≠ yj for each j. We give conditions on the set of d-tuples ða1; : : : ; adÞ so that, for every coloring f : L → 1⁄2c , the grid L 1⁄4 1⁄2a1 × · · · ×1⁄2ad contains a box on which f is constant. In particular, we analyze the set of grids that are minimal with respect to this property.We show that, for d ≥ 3, this set has size Oðcð17·3−1Þ∕ 2Þ, and all its elements have volume Oðcð3−1Þ ∕ 2Þ as c → ∞.