## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Tuesday, November 22, 2005, 12:15 pm

Duration: This information is not available in the database

Location: This information is not available in the database

Speaker: Graham Brightwell (London School of Economics)

## Non-transitive sets of dice

It is well-known that there is a trio of "three-sided dice" A, B and C, with the following property. If A and B are rolled, then the probability that the number showing on die A is greater than that on die B is strictly greater than 1/2 -- A beats B -- while similarly B beats C and C beats A. There is also a set of seven three-sided dice such that, for any pair of them, there is a third that beats both. However, we show that, no matter how many three-sided dice we have, there will always be three dice in the collection that cannot be simultaneously beaten -- a "dominating set" of size three. Generally, we investigate the relationship between the number of sides on the dice and the minimum size of a dominating set. The tools used are mostly combinatorial, with a little elementary game theory and some combinatorial geometry thrown in.

(Joint work with Noga Alon, Hal Kierstead, Sasha Kostochka and Peter Winkler.)

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