Erdős Problem #911

Let \(\hat{R}(G)\) denote the size Ramsey number, the minimal number of edges \(m\) such that there is a graph \(H\) with \(m\) edges that is Ramsey for \(G\). Is there a function \(f\) such that \(f(x)/x\to \infty\) as \(x\to \infty\) such that, for all large \(C\), if \(G\) is a graph with \(n\) vertices and \(e\geq Cn\) edges then \[\hat{R}(G) > f(C) e?\] Here \(\hat{R}(G)\) is `sizeRamsey G G`, and \(G\) has at least one vertex (for \(n = 0\) the inequality fails trivially).