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- W4287180018 abstract "A famous result by ErdH{o}s and Szekeres (1935) asserts that, for all $k,d in mathbb{N}$, there is a smallest integer $n = g^{(d)}(k)$ such that every set of at least $n$ points in $mathbb{R}^d$ in general position contains a $k$-gon, that is, a subset of $k$ points which is in convex position. In this article, we present a SAT model based on acyclic chirotopes (oriented matroids) to investigate ErdH{o}s--Szekeres numbers in small dimensions. To solve the SAT instances we use modern SAT solvers and all our unsatisfiability results are verified using DRAT certificates. We show $g^{(3)}(7) = 13$, $g^{(4)}(8) le 13$, and $g^{(5)}(9) le 13$, which are the first improvements for decades. For the setting of $k$-holes (i.e., $k$-gons with no other points in the convex hull), where $h^{(d)}(k)$ denotes the minimum number $n$ such that every set of at least $n$ points in $mathbb{R}^d$ in general position contains a $k$-hole, we show $h^{(3)}(7) le 14$, $h^{(4)}(8) le 13$, and $h^{(5)}(9) le 13$. Moreover, all obtained bounds are sharp in the setting of acyclic chirotopes and we conjecture them to be sharp also in the original setting of point sets. As a byproduct, we verify previously known bounds. In particular, we present the first computer-assisted proof of the upper bound $h^{(2)}(6)le g^{(2)}(9) le 1717$ by Gerken (2008)." @default.
- W4287180018 created "2022-07-25" @default.
- W4287180018 creator A5078050536 @default.
- W4287180018 date "2021-05-18" @default.
- W4287180018 modified "2023-10-16" @default.
- W4287180018 title "A SAT attack on higher dimensional ErdH{o}s--Szekeres numbers" @default.
- W4287180018 doi "https://doi.org/10.48550/arxiv.2105.08406" @default.
- W4287180018 hasPublicationYear "2021" @default.
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