On Schur properties of random subsets of integers
type: Item
instance of: scholarly article
title: On Schur properties of random subsets of integers
zbMATH Open document ID: 0880.05081
author: Ron Graham
author: Vojtěch Rödl
author: Andrzej Ruciński
published in: Journal of Number Theory
publication date: 1998-01-14T00:00:00Z
review text: A celebrated theorem of Schur asserts that for each \(r>1\) and for \(n>n(r)\), if the set \(\{1,2,\dots,n\}\) is partitioned into \(r\) classes then one of them contains a solution to the equation \(x+y=z\). It is said to be a monochromatic solution. Any solution with \(x\neq y\) is called a Schur triple. A set \(A\subseteq\{1,\dots,n\}\) has the Schur property if for any 2-partition of \(A\) there is a monochromatic Schur triple. The authors prove that \(\sqrt n\) is a threshold for the Schur property showing that for every \(\omega(n)\to\infty\), almost all sets \(A\subseteq\{1,\dots,n\}\) with \(|A|>\omega(n)\sqrt n\) have the Schur property, while almost all \(A\) with \(|A|<\sqrt n/\omega(n)\) do not have this property.
Mathematics Subject Classification ID: 05D10
Mathematics Subject Classification ID: 11B75
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 05A17
zbMATH DE Number: 980276
zbMATH Keywords: theorem of Schur
zbMATH Keywords: monochromatic solution
zbMATH Keywords: Schur triple
zbMATH Keywords: Schur property
zbMATH Keywords: 2-partition
reviewed by: Norbert Hegyvári
MaRDI profile type: Publication
OpenAlex ID: W2103394264
Wikidata QID: Q97694789
DOI: 10.1006/JNTH.1996.0155
description: scientific article; zbMATH DE number 980276