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- W4300477629 abstract "Consider a game played on the edge set of the infinite clique by two players, Builder and Painter. In each round, Builder chooses an edge and Painter colours it red or blue. Builder wins by creating either a red copy of $G$ or a blue copy of $H$ for some fixed graphs $G$ and $H$. The minimum number of rounds within which Builder can win, assuming both players play perfectly, is the on-line Ramsey number $tilde{r}(G,H)$. In this paper, we consider the case where $G$ is a path $P_k$. We prove that $tilde{r}(P_3, P_{ell+1}) = lceil 5ell/4 rceil = tilde{r}(P_3, C_ell)$ for all $ell ge 5$, and determine $tilde{r}(P_4, P_{ell+1}$) up to an additive constant for all $ell ge 3$. We also prove some general lower bounds for on-line Ramsey numbers of the form $tilde{r}(P_{k+1},H)$." @default.
- W4300477629 created "2022-10-03" @default.
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- W4300477629 date "2014-04-15" @default.
- W4300477629 modified "2023-09-23" @default.
- W4300477629 title "On-line Ramsey numbers of paths and cycles" @default.
- W4300477629 doi "https://doi.org/10.48550/arxiv.1404.3894" @default.
- W4300477629 hasPublicationYear "2014" @default.
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