Size Ramsey number of dense graphs: is $\hat R(G)$ superlinear in the edge count? (Erdős #911)

For a graph $G$, let $\hat{R}(G)$ denote the size Ramsey number: the least number of edges $m$ of a graph $H$ that is Ramsey for $G$, meaning every red/blue colouring of the edges of $H$ contains a monochromatic copy of $G$. Is there a function $f$ with $f(x)/x\to\infty$ as $x\to\infty$ such that, for all sufficiently large constants $C$, every graph $G$ with $n$ vertices and $e\geq Cn$ edges satisfies $$\hat{R}(G) > f(C)\,e\,?$$ In words: for graphs whose edge count $e$ is at least linear in the number of vertices, must the size Ramsey number exceed $e$ by a factor that itself grows without bound as the edge-density constant $C$ increases?

Acceptance. FULLY RESOLVES: a complete proof establishing the existence of a function $f$ with $f(x)/x\to\infty$ for which every graph $G$ with $n$ vertices and $e\geq Cn$ edges obeys $\hat{R}(G)>f(C)\,e$ (dense graphs have size Ramsey number superlinear in their edge count, in the stated uniform sense), OR a disproof — an explicit family of graphs $G$ with $e\geq Cn$ edges and $\hat{R}(G)/e$ bounded as $C\to\infty$, together with proof. A machine-checkable proof (Lean/Coq) is preferred, otherwise a full written proof. ADVANCES: prove a nontrivial superlinear lower bound $\hat{R}(G)\gg e\cdot g(C)$ with $g$ unbounded for a natural dense family (e.g. complete graphs or $C$-regular graphs), or an upper-bound construction bounding $\hat{R}(G)/e$ for dense $G$ — each with complete proof; or an exact or near-exact determination of $\hat{R}(G)$ for specific small dense $G$ that establishes new data points, with a reproducible certificate. Deliver the proof file, or the construction plus verification.