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An online Ramsey game (G,H) is a game between Builder and Painter, alternating in turns. During each turn, Builder draws an edge, and Painter colors it blue or red. Builder’s goal is to force Painter to create a monochromatic copy of G, while Painter’s goal is to prevent this. The only limitation for Builder is that after each of his moves, the resulting graph has to belong to the class of grap...

We study on-line version of size-Ramsey numbers of graphs defined via a game played between Builder and Painter: in one round Builder joins two vertices by an edge and Painter paints it red or blue. The goal of Builder is to force Painter to create a monochromatic copy of a fixed graph H in as few rounds as possible. The minimum number of rounds (assuming both players play perfectly) is the on-...