"Integer-making" theorems

József Beck; Tibor Fiala

doi:10.1016/0166-218X(81)90022-6

"Integer-making" theorems

József Beck, Tibor Fiala

Research output: Contribution to journal › Article › peer-review

165 Scopus citations

Abstract

Let X = {x₁, x₂,...} be a finite set and associate to every x_i a real number α_i. Let f(n) [g (n)] be the least value such that given any family F of subsets of X having maximum degree n [cardinality n], one can find integers α_i, i=1,2,... so that α_i - α_i|<1 and ∑ x_{i ε{lunate} E}a_i- ∑ x_{i ε{lunate} E}α_i≤f{hook}(n) ∑ x_{i ε{lunate} E}a_i- ∑ x_{i ε{lunate} E}α_i≤g(n) for all E ε{lunate} F. We prove f(n)≤n - 1 and g(n)≤c(n log n)^{1 2}.

Original language	English (US)
Pages (from-to)	1-8
Number of pages	8
Journal	Discrete Applied Mathematics
Volume	3
Issue number	1
DOIs	https://doi.org/10.1016/0166-218X(81)90022-6
State	Published - Feb 1981
Externally published	Yes

All Science Journal Classification (ASJC) codes

Discrete Mathematics and Combinatorics
Applied Mathematics

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title = "{"}Integer-making{"} theorems",

abstract = "Let X = {x1, x2,...} be a finite set and associate to every xi a real number αi. Let f(n) [g (n)] be the least value such that given any family F of subsets of X having maximum degree n [cardinality n], one can find integers αi, i=1,2,... so that αi - αi|<1 and ∑ xi ε{lunate} Eai- ∑ xi ε{lunate} Eαi≤f{hook}(n) ∑ xi ε{lunate} Eai- ∑ xi ε{lunate} Eαi≤g(n) for all E ε{lunate} F. We prove f(n)≤n - 1 and g(n)≤c(n log n) 1 2.",

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AU - Beck, József

AU - Fiala, Tibor

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N2 - Let X = {x1, x2,...} be a finite set and associate to every xi a real number αi. Let f(n) [g (n)] be the least value such that given any family F of subsets of X having maximum degree n [cardinality n], one can find integers αi, i=1,2,... so that αi - αi|<1 and ∑ xi ε{lunate} Eai- ∑ xi ε{lunate} Eαi≤f{hook}(n) ∑ xi ε{lunate} Eai- ∑ xi ε{lunate} Eαi≤g(n) for all E ε{lunate} F. We prove f(n)≤n - 1 and g(n)≤c(n log n) 1 2.

AB - Let X = {x1, x2,...} be a finite set and associate to every xi a real number αi. Let f(n) [g (n)] be the least value such that given any family F of subsets of X having maximum degree n [cardinality n], one can find integers αi, i=1,2,... so that αi - αi|<1 and ∑ xi ε{lunate} Eai- ∑ xi ε{lunate} Eαi≤f{hook}(n) ∑ xi ε{lunate} Eai- ∑ xi ε{lunate} Eαi≤g(n) for all E ε{lunate} F. We prove f(n)≤n - 1 and g(n)≤c(n log n) 1 2.

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"Integer-making" theorems

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