Online balanced graph avoidance games
European Journal of Combinatorics
Avoiding small subgraphs in Achlioptas processes
Random Structures & Algorithms - Proceedings of the Thirteenth International Conference “Random Structures and Algorithms” held May 28–June 1, 2007, Tel Aviv, Israel
Online ramsey games in random graphs
Combinatorics, Probability and Computing
Offline thresholds for Ramsey-type games on random graphs
Random Structures & Algorithms - Special 20th Anniversary Issue
Small subgraphs in random graphs and the power of multiple choices
Journal of Combinatorial Theory Series B
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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.