  Given a class $\mathcal{C}$ of graphs and a fixed graph $H$, the online
Ramsey game for $H$ on $\mathcal C$ is a game between two players Builder and
Painter as follows: an unbounded set of vertices is given as an initial state,
and on each turn Builder introduces a new edge with the constraint that the
resulting graph must be in $\mathcal C$, and Painter colors the new edge either
red or blue. Builder wins the game if Painter is forced to make a monochromatic
copy of $H$ at some point in the game. Otherwise, Painter can avoid creating a
monochromatic copy of $H$ forever, and we say Painter wins the game.
  We initiate the study of characterizing the graphs $F$ such that for a given
graph $H$, Painter wins the online Ramsey game for $H$ on $F$-free graphs. We
characterize all graphs $F$ such that Painter wins the online Ramsey game for
$C_3$ on the class of $F$-free graphs, except when $F$ is one particular graph.
We also show that Painter wins the online Ramsey game for $C_3$ on the class of
$K_4$-minor-free graphs, extending a result by Grytczuk, Ha{\l}uszczak, and
Kierstead.
