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- W3160770462 abstract "A famous result by Erdős and Szekeres (1935) asserts that, for every (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, i.e., a subset of k points which is in convex position. We present a SAT model for higher dimensional point sets which is based on chirotopes, and use modern SAT solvers to investigate Erdős–Szekeres numbers in dimensions (d=3,4,5). We show (g^{(3)}(7) le 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 chirotopes and we conjecture them to be sharp also in the original setting of point sets." @default.
- W3160770462 created "2021-05-24" @default.
- W3160770462 creator A5078050536 @default.
- W3160770462 date "2021-01-01" @default.
- W3160770462 modified "2023-09-30" @default.
- W3160770462 title "A SAT Attack on Higher Dimensional Erdős–Szekeres Numbers" @default.
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- W3160770462 doi "https://doi.org/10.1007/978-3-030-83823-2_17" @default.
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