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Upper Bounds for 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

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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 an upper bound on the typical duration of this game if F is from a large class of graphs including cliques and cycles of arbitrary size. Together with lower bounds published elsewhere, explicit threshold functions follow.

References

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