论文标题
切断的角落
Cutting corners
论文作者
论文摘要
我们说，如果有$ε> 0 $和$ n_0 $，则$ \ mathbb r^n $的子集$ m $是指数级的。 $ \ mathbb r^n $，因此没有$ m $的副本是单色的。 Euclidean Ramsey理论的一个重要结果是由于Frankl和Rödl造成的，并指出以下内容（在某些温和的额外条件下）：如果$ n_1 $和$ n_2 $均为指定性的Ramsey，那么$ n_1 n_1 \ times n_2 $。该结果多次应用于两点集，这意味着“超矩形”的任何子集都是指数的拉姆西。 但是，通常，这种“嵌入”会导致上述$ε$的效率非常低。在本文中，我们提出了另一种结合指数性的拉姆西集合的方式，在某些重要情况下，这给出了更好的估计。特别是，我们表明，具有禁止等边三角形的$ \ mathbb r^n $的色数满足$χ（\ mathbb r^n，\ triangle）\ ge \ big big（1.0742 ...+o（1）\ big）^n $，大大改善了先前常量的常数$ 1.01444 $。对于较大维度的常规简单以及曼哈顿规范中相关的几何拉姆西型问题，我们还获得了类似的强大结果。 然后，我们证明相同的技术意味着在其他组合问题中的几种有趣的推论。特别是，我们给出了$ \ Mathcal f \ subset2^{[n]} $的大小上的明确上限，其中不包含弱$ k $ -sunflowers，即没有收集$ k $的集合，这些集合与同一大小的成对相交的相交相交。这一界限可改善所有$ k \ ge4 $的先前已知结果。最后，我们还从弗兰克尔和威尔逊的早期结果中简单地扣除了（其他）著名的Frankl-Rödl定理。它可能提供了最短的弗兰克（Frankl）和rödl结果证明，其范围最有效。
We say that a subset $M$ of $\mathbb R^n$ is exponentially Ramsey if there are $ε>0$ and $n_0$ such that $χ(\mathbb R^n,M)\ge(1+ε)^n$ for any $n>n_0$, where $χ(\mathbb R^n,M)$ stands for the minimum number of colors in a coloring of $\mathbb R^n$ such that no copy of $M$ is monochromatic. One important result in Euclidean Ramsey theory is due to Frankl and Rödl, and states the following (under some mild extra conditions): if both $N_1$ and $N_2$ are exponentially Ramsey then so is $N_1\times N_2$. Applied several times to two-point sets, this result implies that any subset of a `hyperrectangle' is exponentially Ramsey. However, generally, such `embeddings' result in very inefficient bounds on the aforementioned $ε$. In this paper, we present another way of combining exponentially Ramsey sets, which gives much better estimates in some important cases. In particular, we show that the chromatic number of $\mathbb R^n$ with a forbidden equilateral triangle satisfies $χ(\mathbb R^n,\triangle)\ge\big(1.0742...+o(1)\big)^n$, greatly improving upon the previous constant $1.0144$. We also obtain similar strong results for regular simplices of larger dimensions, as well as for related geometric Ramsey-type questions in Manhattan norm. We then show that the same technique implies several interesting corollaries in other combinatorial problems. In particular, we give an explicit upper bound on the size of a family $\mathcal F\subset2^{[n]}$ that contains no weak $k$-sunflowers, i.e. no collection of $k$ sets with pairwise intersections of the same size. This bound improves upon previously known results for all $k\ge4$. Finally, we also present a simple deduction of the (other) celebrated Frankl--Rödl theorem from an earlier result of Frankl and Wilson. It gives probably the shortest known proof of Frankl and Rödl result with the most efficient bounds.
