Cutting corners
Andrey Kupavskii, Arsenii Sagdeev, Dmitrii Zakharov

TL;DR
This paper introduces a new method for combining exponentially Ramsey sets that yields significantly improved bounds on the chromatic number of Euclidean spaces avoiding certain geometric configurations, with applications to combinatorial set families.
Contribution
It presents a novel technique for combining exponentially Ramsey sets, leading to better estimates in geometric and combinatorial Ramsey problems, including improved bounds for coloring Euclidean space and sunflower-free set families.
Findings
Chromatic number of space avoiding equilateral triangles pprox (1.0742)^n
Enhanced bounds for regular simplices and Manhattan norm problems
Improved upper bounds on weak sunflower-free families for all k
Abstract
We say that a subset of is exponentially Ramsey if there are and such that for any , where stands for the minimum number of colors in a coloring of such that no copy of 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 and are exponentially Ramsey then so is . 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 . In this paper, we present another way of combining exponentially Ramsey sets, which gives much better estimates in some important cases. In…
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