Difference sets in Quadratic Density Hales Jewett conjecture with 2 letters

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.