Infinitely many minimally non-Ramsey size-linear graphs

A graph $G$ is said to be Ramsey size-linear if $r(G,H)=O_G(e(H))$ for every graph $H$ with no isolated vertices. Erdős, Faudree, Rousseau, and Schelp observed that $K_4$ is not Ramsey size-linear, but each of its proper subgraphs is, and they asked whether there exist infinitely many such graphs. In this short note, we answer this question in the affirmative.