Root systems and the Erdős-Szekeres Problem
نویسندگان
چکیده
منابع مشابه
Erdős - Szekeres Results for Set Partitions
We prove a Ramsey-theoretic result on set partitions of finite sets and a refinement based on the number of blocks in the set partition. A well-known bijection shows that our results are equivalent to results on finite sequences in the spirit of the Erdős-Szekeres theorem. 1. Background and Definitions In their early work on Ramsey theory, Erdős and Szekeres established the following result: Th...
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Let ES(n) denote the least integer such that among any ES(n) points in general position in the plane there are always n in convex position. In 1935, P. Erdős and G. Szekeres showed that ES(n) exists and ES(n) ≤ ` 2n−4 n−2 ́ + 1. Six decades later, the upper bound was slightly improved by Chung and Graham, a few months later it was further improved by Kleitman and Pachter, and another few months ...
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A classical conjecture of Erdős and Szekeres states that, for every integer k ≥ 2, every set of 2k−2 + 1 points in the plane in general position contains k points in convex position. In 2006, Peters and Szekeres introduced the following stronger conjecture: every redblue coloring of the edges of the ordered complete 3-uniform hypergraph on 2k−2 + 1 vertices contains an ordered subhypergraph wit...
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According to the Erdős discrepancy conjecture, for any infinite ±1 sequence, there exists a homogeneous arithmetic progression of unbounded discrepancy. In other words, for any ±1 sequence (x1, x2, ...) and a discrepancy C, there exist integers m and d such that | ∑ m i=1 xi·d| > C. This is an 80-year-old open problem and recent development proved that this conjecture is true for discrepancies ...
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 1997
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa-81-3-229-245