On Pisier Type Theorems

IF 1 2区 数学 Q1 MATHEMATICS Combinatorica Pub Date : 2024-07-11 DOI:10.1007/s00493-024-00115-1
Jaroslav Nešetřil, Vojtěch Rödl, Marcelo Sales
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Abstract

For any integer \(h\geqslant 2\), a set of integers \(B=\{b_i\}_{i\in I}\) is a \(B_h\)-set if all h-sums \(b_{i_1}+\ldots +b_{i_h}\) with \(i_1<\ldots <i_h\) are distinct. Answering a question of Alon and Erdős [2], for every \(h\geqslant 2\) we construct a set of integers X which is not a union of finitely many \(B_h\)-sets, yet any finite subset \(Y\subseteq X\) contains an \(B_h\)-set Z with \(|Z|\geqslant \varepsilon |Y|\), where \(\varepsilon :=\varepsilon (h)\). We also discuss questions related to a problem of Pisier about the existence of a set A with similar properties when replacing \(B_h\)-sets by the requirement that all finite sums \(\sum _{j\in J}b_j\) are distinct.

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论皮西埃类型定理
对于任意整数 \(h\geqslant 2\), 如果所有与 \(i_1<\ldots <i_h\) 的 h-sums \(b_{i_1}+\ldots +b_{i_h})都是不同的,那么整数集合 \(B=\{b_i\}_{i\in I}\) 就是一个 \(B_h\)-set 。为了回答阿隆和厄尔多斯的一个问题[2],对于每一个 \(h\geqslant 2\) 我们都要构造一个整数集合 X,这个集合不是有限多个 \(B_h\)-set 的联合,然而任何有限子集 \(Y\subseteq X\) 都包含一个 \(B_h\)-set Z,其中 \(|Z|geqslant \varepsilon |Y/|),这里 \(\varepsilon :=\varepsilon (h)\).我们还讨论了与皮西埃的一个问题有关的问题,即当所有有限和 \(\sum _{j\in J}b_j\) 都是不同的要求取代 \(B_h\)-set 时,是否存在一个具有类似性质的集合 A。
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来源期刊
Combinatorica
Combinatorica 数学-数学
CiteScore
1.90
自引率
0.00%
发文量
45
审稿时长
>12 weeks
期刊介绍: COMBINATORICA publishes research papers in English in a variety of areas of combinatorics and the theory of computing, with particular emphasis on general techniques and unifying principles. Typical but not exclusive topics covered by COMBINATORICA are - Combinatorial structures (graphs, hypergraphs, matroids, designs, permutation groups). - Combinatorial optimization. - Combinatorial aspects of geometry and number theory. - Algorithms in combinatorics and related fields. - Computational complexity theory. - Randomization and explicit construction in combinatorics and algorithms.
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