Monochromatic combinatorial lines of length three

TL;DR

This paper proves that for sufficiently many colors growing with the double logarithm of the dimension, any coloring of the cube contains a monochromatic combinatorial line, extending understanding of combinatorial structures in high dimensions.

Contribution

It establishes a bound on the number of colors needed to guarantee a monochromatic combinatorial line in high-dimensional cubes, with the number of colors growing as a constant times log log n.

Findings

Existence of a constant c for monochromatic lines in colored cubes

Colorings with c log log n colors always contain a monochromatic line

Advances the understanding of combinatorial lines in high-dimensional spaces

Abstract

We show that there is a constant $c$ such that any colouring of the cube $[3]^n$ in $c \log \log n$ colours contains a monochromatic combinatorial line.