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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 759.05095

**Autor: ** Erdös, Paul; Galvin, Fred

**Title: ** Some Ramsey-type theorems. (In English)

**Source: ** Discrete Math. 87, No.3, 261-269 (1991).

**Review: ** For any set A let [A]^{r} denote the collection of r-element subsets of A. By a k-coloring of the r-subsets of A we mean a function f: [A]^{r} ––> **{**1,...,k**}**. A set X\subset A is said to be f-homogeneous if f is a constant on [X]^{r}. The partition symbol a ––> (x)^{r}_{k} denotes the assertion: given a set A with | A| = a and a coloring f: [A]^{r} ––> **{**1,...,k**}**, there is an f- homogeneous set X\subset A with |X| \leq x.

The main result of this paper is

Theorem 2.1. Let r and k be positive integers, and let the function \phi: **N** ––> **R** be such that n ––> (\phi(n))^{r}_{k+1} holds for all sufficienty large n. Given any coloring f: [**N**]^{r} ––> **{**1,...,k**}**, there is a set A\subset**N** such that: (1) |**{**f(X): X in [A]^{r}**}**| \leq 2^{r-1}; (2) |A\cap**{**1,...,n**}**| \geq \phi(n) for infinitely many n.

**Reviewer: ** J.E.Graver (Syracuse)

**Classif.: ** * 05D10 Ramsey theory

**Keywords: ** Ramsey-type theorems; homogeneous set; partition; coloring

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