On Pisier Type Theorems

For any integer h ⩾ 2 , a set of integers B = { b i } i ∈ I is a B h -set if all h -sums b i 1 + … + b i h with i 1 < … < i h are distinct. Answering a question of Alon and Erdős [ 2 ], for every h ⩾ 2 we construct a set of integers X which is not a union of finitely many B h -sets, yet any fi...

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Bibliographic Details
Published in:Combinatorica (Budapest. 1981) 2024-12, Vol.44 (6), p.1211-1232
Main Authors: Nešetřil, Jaroslav, Rödl, Vojtěch, Sales, Marcelo
Format: Article
Language:English
Subjects:
Combinatorics
Integers
Mathematics
Mathematics and Statistics
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Summary:For any integer h ⩾ 2 , a set of integers B = { b i } i ∈ I is a B h -set if all h -sums b i 1 + … + b i h with i 1 < … < i h are distinct. Answering a question of Alon and Erdős [ 2 ], for every h ⩾ 2 we construct a set of integers X which is not a union of finitely many B h -sets, yet any finite subset Y ⊆ X contains an B h -set Z with | Z | ⩾ ε | Y | , where ε : = ε ( 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 ∑ j ∈ J b j are distinct.
ISSN:0209-9683
1439-6912
DOI:10.1007/s00493-024-00115-1