Allen–Brightwell–Skokan cycle goodness conjecture

For a graph $H$, let $\chi(H)$ denote its chromatic number and $|H|$ its number of vertices. A graph $G$ is $H$-good if $R(G,H)=\max\{(|G|-1)(\chi(H)-1)+\sigma(H),(|H|-1)(\chi(G)-1)+\sigma(G)\}$, where $\sigma(F)$ is the minimum size of a color class in a proper $\chi(F)$-coloring of $F$. Allen–Brightwell–Skokan's conjecture. For any graph $H$ and $m\geq \chi(H)|H|$, the cycle $C_m$ is $H$-good. The conjecture would extend the known theory of Ramsey goodness to cycles against arbitrary graphs; the source presents it as open, including the cases relevant to $H=K_{2,n}$.