Generalizations of Schütte’s property with applications to dice games
Date
2024-08
Authors
Jeffries, Joel Andrew
Major Professor
Advisor
Butler, Steve
Lidický, Bernard
Zerbib, Shira
Lutz, Jack
Song, Sung-Yell
Committee Member
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Abstract
In this work, we explore dice games of the following kind: k players choose a die from a
collection of dice, and one final player chooses a die in response. The players then roll their dice
and compare the results. Traditionally, these games have been represented by tournaments.
Tournaments that give the last player an advantage are related to Sch ̈utte’s property, introduced
by Erd ̋os in 1963. A tournament has Sch ̈utte’s property if for every k-set of vertices, there is
another vertex directed toward that k-set. We examine various modifications of this property and
the dice games associated with them.
In Chapters 2 and 3, we provide background on Sch ̈utte’s property and dice sets that can
realize a given tournament. We also explore computational techniques for searching for such
tournaments and sets of dice.
In Chapter 4, inspired by Grime’s dice, we generalize Sch ̈utte’s property to sets of tournaments
and study the minimum number of vertices needed to form such sets. Additionally, we consider
methods for realizing these sets with dice by rolling them multiple times and summing the results.
In Chapter 5, we extend the concept of Sch ̈utte’s property to directed hypergraphs. We
determine the exact values for the minimum order of hypergraphs with this property.
In Chapter 6, we explore how to realize the relationships in such hypergraphs using dice. We
demonstrate that any ordering of outcomes in three-wise rolls among n players can be achieved
with a set of dice.
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dissertation