Cutting corners

TL;DR

This work advances Euclidean and Manhattan Ramsey theory by developing a tree-like concatenation framework that improves how Ramsey-configurations are composed, replacing earlier product-based constructions. The authors prove a Frankl–Rödl product theorem tailored to their framework and apply it to obtain substantially stronger exponential lower bounds, notably showing $\chi(\mathbb{R}^n, \triangle) \ge (1.0742...+o(1))^n$ and, more generally, $\chi(\mathbb{R}^n, \triangle^k) \ge (\psi_2^{1/(k+1)}+o(1))^n$. They also derive explicit bounds in Manhattan geometry, e.g., $\chi(\mathbb{R}_1^n, \triangle^k)$, and obtain corollaries for weak sunflowers, including an upper bound $|\mathcal{F}| \le (\delta(\rho,\sigma)+o(1))^{n}$ in the $s$-avoiding regime and improved constants for related intersection problems. The paper connects these geometricRamsey results to combinatorial set families, culminating in a concise deduction of Frankl–Rödl from Frankl–Wilson. Overall, the methods yield sharper constants and broaden applicability to both Euclidean and Manhattan settings, with several open problems guiding future work.

Abstract

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.

Theorems & Definitions (17)

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• Lemma 1