An improved lower bound for Folkman's theorem

Ramsey type of results in additive combinatorics could be originated to Hilbert. In this talk a brief history of the topics and some recent developments will be explained. Folkman's Theorem asserts that for each k in N, there exists a natural number n=F(k) such that whenever the elements of [n] are two-coloured, there exists a subset A of [n] of size k with the property that all the sums of the form \sum_{x\in B} x, where B is a nonempty subset of A, are contained in [n] and have the same colour. In 1989, Erdős and Spencer showed that F(k) >= 2^{ck^2/log k}, where c>0 is an absolute constant; here, we improve this bound.