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**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 = **{**x_{i}**}** is simple if all gaps x_{i+1}-x_{i} are different. Two simple sequences x and y are distance similar if the consecutive distances are ordered in the same way, that is x_{i+1}-x_{i} < x_{j+1}-x_{j} iff y_{i+1}-y_{i} < y_{j+1}-y_{j} 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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