Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 884.05092
Autor: Erdös, Paul; Hajnal, A.; Pach, János
Title: On a metric generalization of Ramsey's theorem. (In English)
Source: Isr. J. Math. 102, 283-295 (1997).
Review: An increasing sequence of reals x = {xi} is simple if all gaps xi+1-xi are different. Two simple sequences x and y are distance similar if the consecutive distances are ordered in the same way, that is xi+1-xi < xj+1-xj iff yi+1-yi < yj+1-yj for all pairs i,j. The paper proves that given any bounded simple sequence x and any colouring of the pairs of rational numbers by finite number of colours, there is always a sequence y distance similar to x such that all pairs of y are of the same colour. A number of analogous results are proved and some interesting counterexamples are given.
Reviewer: P.L.Erdös (Budapest)
Classif.: * 05D10 Ramsey theory
Keywords: Ramsey's theory; Szemerédi's theorem; partition calculus
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