An Analogue of Gromov's Waist Theorem for Coloring the Cube

Introduction One possible way to express that the cube Q d = [0, 1] d has dimension d is to notice that it cannot be colored in d colors with arbitrarily small connected monochromatic components. A more precise statement was proved by Lebesgue: If the unit cube Q d is covered by closed sets with multiplicity at most d then one covering set must meet two opposite facets of Q d. An analogue of Gromov's waist theorem for coloring the cube July 2013 2 / 27 Introduction Matoušek and Přív˘ etiv´y (2008) asked a similar question in combinatorial setting: Question If we color Q d in m + 1 colors (to make the problem discrete we color small cubes of the partition of Q d into n d small cubes) then what size of a monochromatic connected component can we guarantee? Introduction The corresponding combinatorial statement for m = d − 1, corresponding to Lebesgue's theorem, is also known as the HEX lemma. In this case there must exist a monochromatic connected component spanning two opposite facets and such a component must consist of at least n small cubes. Matoušek and Přív˘ etiv´y considered colorings in 2 colors using isoperimetric inequalities for the grid and a lower bound n d−1 − d 2 n d−2 for the size of a connected monochromatic component was established. They also conjectured that the size of a monochromatic connected component is of order n d−m for m + 1 colors when d and m are fixed. Introduction Alexey Kanel-Belov also posed the same problem in 1990s (as he told me) and it circulated among mathematicians in Moscow and was posed at some olympiad-like events, in particular at the meeting of the Tournament of Towns 2010. After discussions in 2010, a proof, different from what we discuss here, was obtained by an undergraduate student Marsel Matdinov, see arXiv:1111.3911 and the final version in DCG (2013). Introduction So we confirm the conjectured lower bound for the size of a monochromatic connected component: Theorem (The main theorem) Suppose that a d-dimensional cube Q d is partitioned into n d small cubes in an obvious way. Let 0 ≤ m < d. If the set of small cubes of Q d is colored in m + 1 colors then there exists a connected monochromatic component of size at least f (d, m)n d−m. Here f (d, m) is …