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. 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∈H 1log h⩾C log log log log nlog log log log log n. nswers positively a conjecture of Erdös (see “On the combinatorial problems which I would most like to see solved”, Combinatorica 1 (1981) 25).