Monochromatic sumsets (Q1116969)

The sumset \(P(S)\) is defined to be the set of all finite sums of distinct elements in \(S\subset\mathbb N\). The number \(F(k)\) is defined to be the least \(n\) such that if \(\{1,\ldots,n\}\) is two coloured then there is a \(k\)-set \(S\) with \(P(S)\subset \{1,\ldots,n\}\) and \(P(S)\) monochromatic. A short proof that \(F(k)>2^{ck^2/\log k}\) is given, and a conjecture related to removing the logarithmic term is posed.