On homogeneous sets of positive integers

AbstractThe 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 minH⩾2 such that∑h∈H1logh⩾Cloglogloglognlogloglogloglogn.This answers positively a conjecture of Erdös (see “On the combinatorial problems which I would most like to see solved”, Combinatorica 1 (1981) 25)