  Consider the balanced Ramsey game, in which a player has r colors and where
in each step r random edges of an initially empty graph on n vertices are
presented. The player has to immediately assign a different color to each edge
and her goal is to avoid creating a monochromatic copy of some fixed graph F
for as long as possible. The Achlioptas game is similar, but the player only
loses when she creates a copy of F in one distinguished color. We show that
there is an infinite family of non-forests F for which the balanced Ramsey game
has a different threshold than the Achlioptas game, settling an open question
by Krivelevich et al. We also consider the natural vertex analogues of both
games and show that their thresholds coincide for all graphs F, in contrast to
our results for the edge case.
