Summary: An online Ramsey game is a game between Builder and Painter, alternating in turns. They are given a fixed graph \(H\) and a an infinite set of independent vertices \(G\). In each round Builder draws a new edge in \(G\) and Painter colors it either red or blue. Builder wins if after some finite round there is a monochromatic copy of the graph \(H\), otherwise Painter wins. The online Ramsey number \(\widetilde{r}(H)\) is the minimum number of rounds such that Builder can force a monochromatic copy of \(H\) in \(G\). This is an analogy to the size-Ramsey number \(\overline{r}(H)\) defined as the minimum number such that there exists graph \(G\) with \(\overline{r}(H)\) edges where for any edge two-coloring \(G\) contains a monochromatic copy of \(H\). In this extended abstract, we introduce the concept of induced online Ramsey numbers: the induced online Ramsey number \(\widetilde{r}_{ind}(H)\) is the minimum number of rounds Builder can force an induced monochromatic copy of \(H\) in \(G\). We prove asymptotically tight bounds on the induced online Ramsey numbers of paths, cycles and two families of trees. Moreover, we provide a result analogous to D. Conlon [SIAM J. Discrete Math. 23, No. 4, 1954–1963 (2009; Zbl 1213.05177)], showing that there is an infinite family of trees \(T_1,T_2,\dots, |T_i|<|T_{i+1}|\) for \(i\ge 1\), such that \[\lim _{i\rightarrow \infty } \frac{\widetilde{r}(T_i)}{\overline{r}(T_i)} = 0.\]
For the entire collection see [Zbl 1416.68013].