Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2020
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2008.08556 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909207935057920 |
|---|---|
| author | Pathak, Aritro |
| author_facet | Pathak, Aritro |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2008_08556 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Difference sets in Quadratic Density Hales Jewett conjecture with 2 letters Pathak, Aritro Combinatorics Dynamical Systems 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. |
| title | Difference sets in Quadratic Density Hales Jewett conjecture with 2 letters |
| topic | Combinatorics Dynamical Systems |
| url | https://arxiv.org/abs/2008.08556 |