Abstract
The (m, n)-online Ramsey game is a combinatorial game between two players, Builder and Painter. Starting from an infinite set of isolated vertices, Builder draws an edge on each turn and Painter immediately paints it red or blue. Builder’s goal is to force Painter to create either a red \(K_m\) or a blue \(K_n\) using as few turns as possible. The online Ramsey number \(\tilde{r}(m,n)\) is the minimum number of edges Builder needs to guarantee a win in the (m, n)-online Ramsey game. By analyzing the special case where Painter plays randomly, we obtain an exponential improvement \(\tilde{r}(n,n) \ge 2^{(2-\sqrt{2})n + O(1)}\) for the lower bound on the diagonal online Ramsey number, as well as a corresponding improvement \(\tilde{r}(m,n) \ge n^{(2-\sqrt{2})m + O(1)}\) for the off-diagonal case, where \(m\ge 3\) is fixed and \(n\rightarrow \infty \). Using a different randomized Painter strategy, we prove that \(\tilde{r}(3,n)=\tilde{\Theta }(n^3)\), determining this function up to a polylogarithmic factor. We also improve the upper bound in the off-diagonal case for \(m \ge 4\). In connection with the online Ramsey game with a random Painter, we study the problem of finding a copy of a target graph H in a sufficiently large unknown Erdős–Rényi random graph G(N, p) using as few queries as possible, where each query reveals whether or not a particular pair of vertices are adjacent. We call this problem the Subgraph Query Problem. We determine the order of the number of queries needed for complete graphs up to five vertices and prove general bounds for this problem.
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Acknowledgements
The first author’s research is supported by a Royal Society University Research Fellowship and by ERC Starting Grant 676632. The second author’s research is supported by a Packard Fellowship and by NSF Career Award DMS-1352121. The fourth author’s research is supported by a NSF Graduate Research Fellowship DGE-1656518. We are extremely grateful to Joel Spencer for pointing out a serious flaw in our previous proof of Theorem 1.4 which had been based on a generalization of the Lovász Local Lemma [15]. In the current version, we have a correct proof using a different approach. We would also like to thank the referee for some helpful remarks and Benny Sudakov for bringing the paper of Krivelevich [22] to our attention.
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© 2019 János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature
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Conlon, D., Fox, J., Grinshpun, A., He, X. (2019). Online Ramsey Numbers and the Subgraph Query Problem. In: Bárány, I., Katona, G., Sali, A. (eds) Building Bridges II. Bolyai Society Mathematical Studies, vol 28. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-59204-5_4
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