Abstract
Bradač, Gishboliner and Sudakov asked whether \(R(H, K_{n,n}) = O(n^2)\) for every \(2\)-degenerate graph \(H\). The same bound is predicted by the question of Erdős, Faudree, Rousseau and Schelp on Ramsey size-linear graphs (Erdős Problem \#566) for graphs in which every \(j\)-vertex subgraph spans at most \(2j-3\) edges. We prove \(R(\widetilde{W}_k, K_{n,n}) \le (8k+16)n^2\) for the wheel \(W_k\) minus a spoke, for every \(k \ge 4\), and \(R(J, K_{n,n}) \le 20n^2\) for a graph \(J\) on eight vertices. For every bipartite graph \(F\) we prove \(R(\widetilde{W}_5, F) = O(e(F)\sqrt{\log e(F)})\), and we also obtain \(R(\widetilde{W}_5, K_n) = O(n^{5/2}/\log^{3/2} n)\). If \(\widetilde{W}_5\) is Ramsey size-linear, then \(W_5\) is an explicit minimally non-Ramsey size-linear graph. The proofs combine minimum-degree cores of graphs whose complement contains no \(K_{n,n}\), double counting of pairs with few common neighbours, and the fact that in a \(\widetilde{W}_5\)-free graph the edges lying in at least four triangles form a \(C_4\)-free graph inside every neighbourhood. We also introduce construction sequences, which give \(R(G, K_{n,n}) = O(n^2)\) for every graph admitting one. A computer search shows that \(38\) of the \(49\) minimal open cases of Problem \#566 on at most eight vertices admit such a sequence.