A note on Ramsey size-linear graphs (Q2778280)

If \(G\) is a Ramsey size-linear graph (i.e., there exists a constant \(C_G\) such that for any graph \(H\), the generalized Ramsey number \(r(G, H)\) is bounded above by \(C_G \cdot\text{size}(H) + \text{order}(H)\)), and \(x\) and \(y\) are vertices of \(G\), then adding a sufficiently long path from \(x\) to \(y\) results in another Ramsey size-linear graph. It follows that for any graph \(G\), if every cycle in \(G\) has at least four consecutive vertices of degree 2, then \(G\) is Ramsey size-linear. Incidentally, it also follows that for any graph \(G\), there is a Ramsey size-linear graph \(G'\) obtained by repeatedly subdividing \(G\)'s edges.