Some Ramsey results for the n-cube

TL;DR

This paper proves a Ramsey-type theorem for subsets of the n-cube and applies it to bounds on combinatorial structures like Hales-Jewett lines, Hilbert cubes, corners, and geometric progressions.

Contribution

It establishes new Ramsey results for the n-cube and derives bounds for various combinatorial and number-theoretic configurations.

Findings

Bounds on partial Hales-Jewett lines for alphabets of size 3 and 4

Estimates for Hilbert cubes in sets with small sumsets

Results on corners in the integer lattice and 3-term geometric progressions

Abstract

In this note we establish a Ramsey-type result for certain subsets of the $n$-dimensional cube. This can then be applied to obtain reasonable bounds on various related structures, such as (partial) Hales-Jewett lines for alphabets of sized 3 and 4, Hilbert cubes in sets of real numbers with small sumsets, "corner" in the integer lattice in the plane, and 3-term geometric progressions in integers.