Difference sets in Quadratic Density Hales Jewett conjecture with 2 letters

TL;DR

This paper proves that dense subsets of binary n^2-dimensional spaces contain pairs of elements differing in a rectangular pattern, advancing understanding of the quadratic density Hales-Jewett conjecture with 2 letters.

Contribution

It provides an elementary quantitative proof that dense subsets contain pairs with difference sets of rectangular form, and presents examples of dense subspaces with restricted wildcard sets.

Findings

Dense subsets contain pairs with difference sets of the form γ₁×γ₂.

Examples of dense subspaces with wildcard sets of restricted size and shape.

Elementary quantitative approach to the quadratic density Hales-Jewett conjecture.

Abstract

The Quadratic Density Hales Jewett conjecture with $2$ letters states that for large enough $n$, every dense subset of $\{0,1\}^{n^{2}}$ contains a combinatorial line where the wildcard set is of the form $\gamma \times \gamma$ where $\gamma \subset \{1,2,\dots n\}$. We show in an elementary quantitative way that every dense subset of $\{0,1\}^{n^{2}}$, for sufficiently large $n$, contains two elements such that the set of coordinate points where they differ, which we term the difference set of these two elements, is of the form $\gamma_{1}\times \gamma_{2}$ where $\gamma_1, \gamma_2$ are both nonempty subsets of $\{1,2,\dots n\}$. Further we give several non-trivial examples of dense vector subspaces of $\{0,1\}^{n^{2}}$, where in each case the wildcard set of the combinatorial line that can be obtained has restrictions on its size and shape.