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⊆[n] (i.e., all pairs of H are contained in one of the two partition classes) with min H ≥ 2 such that ∑h∈H1\logh ≥ 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).