Abstract
An online Ramsey game is a game between Builder and Painter, alternating in turns.
They are given a graph and a graph 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 , otherwise Painter wins.
The online Ramsey number is the minimum number of rounds such that Builder can force a monochromatic copy of in .
This is an analogy to the size-Ramsey number defined as the minimum number such that there exists graph with edges where for any edge two-coloring contains a monochromatic copy of .
In this paper, we introduce the concept of induced online Ramsey numbers:
the induced online Ramsey number is the minimum number of rounds Builder can force an induced monochromatic copy of in .
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 , for , such that