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Online Ramsey Games in Random Graphs

Published online by Cambridge University Press:  01 March 2009

MARTIN MARCINISZYN ,
RETO SPÖHEL  and
ANGELIKA STEGER
MARTIN MARCINISZYN
Affiliation:
Institute of Theoretical Computer Science, ETH Zürich, 8092 Zürich, Switzerland (e-mail: mmarcini@inf.ethz.ch, rspoehel@inf.ethz.ch, steger@inf.ethz.ch)
RETO SPÖHEL
Affiliation:
Institute of Theoretical Computer Science, ETH Zürich, 8092 Zürich, Switzerland (e-mail: mmarcini@inf.ethz.ch, rspoehel@inf.ethz.ch, steger@inf.ethz.ch)
ANGELIKA STEGER
Affiliation:
Institute of Theoretical Computer Science, ETH Zürich, 8092 Zürich, Switzerland (e-mail: mmarcini@inf.ethz.ch, rspoehel@inf.ethz.ch, steger@inf.ethz.ch)
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Abstract

Consider the following one-player game. Starting with the empty graph on n vertices, in every step a new edge is drawn uniformly at random and inserted into the current graph. This edge has to be coloured immediately with one of r available colours. The player's goal is to avoid creating a monochromatic copy of some fixed graph F for as long as possible. We prove a lower bound of nβ(F,r) on the typical duration of this game, where β(F,r) is a function that is strictly increasing in r and satisfies limr→∞ β(F,r) = 2 − 1/m2(F), where n2−1/m2(F) is the threshold of the corresponding offline colouring problem.

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