Online Ramsey numbers: long versus short cycles (Q6654127)

\textit{P. Erdős} et al. [Period. Math. Hung. 9, 145--161 (1977; Zbl 0331.05122)] introduced the size-Ramsey number \(\hat{r}(H_1,H_2)\) of a pair of graphs \(H_1\), \(H_2\). Let \(m\) be the least integer such that there exists an \(m\)-edge graph \(G\) with the property that every two-edge coloring of \(G\) results in a monochromatic copy of \(H_i\) in the \(i\)th color. The authors here probe the game variant of the size-Ramsey number called the online size-Ramsey number. An online Ramsey game is played between Builder and Painter on an infinite board \(K_N\), a complete graph on \(N\) vertices. In every round Builder selects an edge, then Painter colors it red or blue. Both know target graphs \(H_1\) and \(H_2\). Builder aims to create either a red copy of \(H_1\) or a blue copy of \(H_2\) in \(K_N\) as soon as possible and Painter tries to prevent it. The online Ramsey number \(\tilde{r}(H_1,H_2)\) is the minimum number of rounds such that the Builder wins. The authors attempt to precisely find the multiplicative constant \(n\) within \(\tilde{r}(C_k,C_n)\) provided that \(k\ge 3\) is fixed. For even \(k\) they found \(\tilde{r}(C_k,C_n)\) up to an additive term depending on \(k\), while for odd \(n\), they found a linear upper bound which is not far from optimal.