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Difference sets in Quadratic Density Hales Jewett conjecture with 2 letters

Aritro Pathak

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 $γ\times γ$ where $γ\subset \{1,2,\dots n\}$. 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 $γ_{1}\times γ_{2}$ where $γ_1, γ_2$ 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.

Difference sets in Quadratic Density Hales Jewett conjecture with 2 letters

Abstract

The Quadratic Density Hales Jewett conjecture with letters states that for large enough , every dense subset of contains a combinatorial line where the wildcard set is of the form where . We show in an elementary quantitative way that every dense subset of , for sufficiently large , 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 where are both nonempty subsets of . Further we give several non-trivial examples of dense vector subspaces of , where in each case the wildcard set of the combinatorial line that can be obtained has restrictions on its size and shape.

Paper Structure

This paper contains 6 sections, 9 theorems.

Table of Contents

  1. Introduction
  2. Symmetric differences and addition in the vector space $\mathbb{F}_{2}^{n^{2}}$
  3. Proof of \ref{['thm:mainthm1']}
  4. Rectangular and square wildcard sets
  5. Further questions.
  6. Acknowledgements

Key Result

Theorem \oldthetheorem

For positive integers $k,r$, there exists a positive integer $N(k,r)$ such that for all $n\geq N(k,r)$ whenever $K[n]:=[k]^{n} \times [k]^{n^{2}}\times \dots \times [k]^{n^{d}}$ is $r$ colored, there exists $a\in K[n]$ and $\gamma\subset [N]$ such that the set of points $\{a\oplus x_1\gamma \oplus x

Theorems & Definitions (15)

  • Conjecture 1: Quadratic Density Hales Jewett with 2 letters
  • Theorem \oldthetheorem: Polynomial Hales Jewett:
  • Theorem \oldthetheorem: Szemeredi:
  • Theorem \oldthetheorem: Density Hales Jewett:
  • Theorem \oldthetheorem: Polynomial van der Waerden:
  • Theorem \oldthetheorem: Polynomial Szemeredi:
  • Conjecture 2: Polynomial Density Hales Jewett
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • ...and 5 more