Cutting corners

We say that a subset $M$ of $\mathbb R^n$ is exponentially Ramsey if there are $\epsilon>0$ and $n_0$ such that $\chi(\mathbb R^n,M)\ge(1+\epsilon)^n$ for any $n>n_0$, where $\chi(\mathbb R^n,M)$ 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 R\"odl, 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 $\epsilon$. 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 $\chi(\mathbb R^n,\triangle)\ge\big(1.0742...+o(1)\big)^n$, 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--R\"odl theorem from an earlier result of Frankl and Wilson. It gives probably the shortest known proof of Frankl and R\"odl result with the most efficient bounds.