On homogeneous sets of positive integers

The main result of this paper is the proof of the following partition property of the family of all two-element sets of the first n positive integers. There is a real constant C>0 such that for every partition of the pairs of the set [n] {1, 2, ..., n} into two parts, there exists a homogeneous set H subset of or equal to [n] (i.e., all pairs of H are contained in one of the two partition classes) with min H greater than or equal to 2 such that Sigma/(h is an element of H) (1)/(log h) greater than or equal to C (log log log log n)/(log log log log log n) This answers positively a conjecture of Erdos (see "On the combinatorial problems which I would most like to see solved", Combinatorica 1 (1981) 25). (C) 2003 Elsevier Science (USA). All rights reserved.