Erdős Problem #570

Let \(k\geq 3\). Is it true that, if \(m\) is sufficiently large, for any graph \(H\) on \(m\) edges without isolated vertices, \[R(C_k,H) \leq 2m+\left\lfloor\frac{k-1}{2}\right\rfloor?\] This was proved for even \(k\) by Erdős, Faudree, Rousseau, and Schelp [EFRS93]. This was proved for \(k=3\) independently by Goddard and Kleitman [GoKl94] and Sidorenko [Si91]. This was proved for \(k=5\) by Jayawardene [Ja99]. Finally it was proved for all odd \(k\geq 7\) by Cambie, Freschi, Morawski, Petrova, and Pokrovskiy [CFMPP26]. This problem is #35 in Ramsey Theory in the graphs problem collection.