An Improved Lower Bound for Folkman’s Theorem

Folkman’s theorem asserts that for each k ∈ N, there exists a natural number n = F (k) such that whenever the elements of [n] are two-coloured, there exists a set A ⊂ [n] of size k with the property that all the sums of the form ∑ x∈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) ≥ 2ck2/ log , where c > 0 is an absolute constant; here, we improve this bound significantly by showing that F (k) ≥ 22/k for all k ∈ N.