On Induced Online Ramsey Number of Paths, Cycles, and Trees
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An online Ramsey game is a game between Builder and Painter, alternating in turns. They are given a graph H and a graph G of an infinite set of independent vertices. In each round Builder draws an edge 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 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 r(H) defined as the minimum number such that there exists graph G with r(H) edges where for any edge two-coloring G contains a monochromatic copy of H. In this paper, we introduce the concept of induced online Ramsey numbers: the induced online Ramsey number 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 Conlon [On-line Ramsey Numbers, SIAM J. Discr. Math. 2009], showing that there is an infinite family of trees T_1,T_2,..., |T_i|<|T_i+1| for i>1, such that _i→∞r(T_i)/r(T_i) = 0.
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