Large Monochromatic Components in Two-Colored Grids

Abstract

Let $D^d_n$ denote the d-dimensional grid with diagonals, that is, the graph with vertex set $\{1,2,\ldots,n\}^d$ and with edges connecting every two vertices that differ by at most 1 in every coordinate. We prove that for an arbitrary coloring of the vertices of $D^d_n$ by two colors there exists a monochromatic connected subgraph with at least $n^{d-1}-d^2n^{d-2}$ vertices; and thus the “horizontal layer” coloring (by the parity of the first coordinate) is almost optimal. We also consider a d-dimensional triangulated grid; this is the graph (1-skeleton) of a triangulation of the solid cube $[1,n]^d$ that refines the subdivision of $[1,n]^d$ into the grid of unit cubes. Here every two-coloring has a monochromatic connected subgraph with $\Omega(n^{d-1}/\sqrt d)$ vertices. These results are proved by combining combinatorial and topological arguments with suitable isoperimetric inequalities, and they can be viewed as d-dimensional generalizations of the planar HEX lemma.