论文标题
二次密度的差异集，哈利斯犹太人猜想，用2个字母
Difference sets in Quadratic Density Hales Jewett conjecture with 2 letters
论文作者
论文摘要
二次密度HALES JEWETT以$ 2 $的字母猜想指出，对于足够大的$ n $，每个密集的子集的$ \ {0,1 \}^{n^{2}} $包含一条组合线，其中通配符集是$γ$γ$γ$γ$γ\umγ\ sebset的形式$γ$γ$γ\ sebset \ spet \ s。 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 $γ_1, γ_2$都是$ \ {1,2，\ dots n \} $的非空子集。此外，我们给出了$ \ {0,1 \}^{n^{2}} $的密集矢量子空间的几个非平地示例，在每种情况下，可以获得的组合线的通配符集都对其大小和形状都有限制。
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.
