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Symbolic-computational methods in combinatorial game theory and Ramsey theory
Descriptive
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Text
TitleInfo
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Title
Symbolic-computational methods in combinatorial game theory and Ramsey theory
Identifier
ETD_1205
Identifier
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http://hdl.rutgers.edu/1782.1/rucore10001600001.ETD.000054841
Language
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eng
Genre
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theses
Subject
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(authority = RUETD)
Topic
Mathematics
Subject
(ID = SBJ-2);
(authority = ETD-LCSH)
Topic
Game theory
Subject
(ID = SBJ-3);
(authority = ETD-LCSH)
Topic
Ramsey theory
Abstract
(type = abstract)
This thesis is a contribution to the emerging fleld of experimental rigorous mathematics, where one uses symbolic computation to conjecture proof-plans, and then proceeds to verify the conjectured proofs rigorously. The proved results, in addition to their independent interest, should also be viewed as case studies in this budding methodology. We now proceed to described the specific results presented in this dissertation. We first develop a finite-state automata approach, implemented in a Maple package ToadsAndFrogs, for conjecturing, and then rigorously proving, values for large families of positions in Richard Guy's combinatorial game Toads and Frogs". In particular, we prove conjectures of Jeff Erickson. We also discuss the values of all positions with exactly one ¤;Ta¤¤Fa;Ta¤¤¤FFF;Ta¤¤Fb, Ta¤¤¤Fb. We next consider the generalized chess problem of checkmating a king with a king and a rook on an m x n board at a specific starting position. We analyze the fastest way to checkmate. We also consider a problem posed by Ronald Graham about the minimum number,
over all 2-colorings of [1; n], of generalized so-called Schur triples, i.e. monochromatic triples of the form (x; y; x + ay) a [greater than or equal to symbol] 1. (The case a = 1 corresponds to the classical Schur triples). In addition to giving a completely new proof of the already known case of a = 1, we show that the minimum number of such triples is at most n2 2a(a2+2a+3) +O(n)
when a [greater than or equal to symbol] 2. We also find a new upper bound for the minimum number, over all r-colorings of [1; n], of monochromatic Schur triples, for r [greater than or equal to symbol] 3. Finally, in yet a different direction, we find closed-form expressions for the second moment of the random variable
umber of monochromatic Schur triples" defined on
the sample space of all r-colorings of the first n integers, and second and even higher moments for the number of monochromatic complete graphs Kk in Kn. In addition to their considerable independent interest, these formulas would hopefully be instrumental
in improving the extremely weak known lower bounds for the asymptotics of Ramsey number.
over all 2-colorings of [1; n], of generalized so-called Schur triples, i.e. monochromatic triples of the form (x; y; x + ay) a [greater than or equal to symbol] 1. (The case a = 1 corresponds to the classical Schur triples). In addition to giving a completely new proof of the already known case of a = 1, we show that the minimum number of such triples is at most n2 2a(a2+2a+3) +O(n)
when a [greater than or equal to symbol] 2. We also find a new upper bound for the minimum number, over all r-colorings of [1; n], of monochromatic Schur triples, for r [greater than or equal to symbol] 3. Finally, in yet a different direction, we find closed-form expressions for the second moment of the random variable
umber of monochromatic Schur triples" defined on
the sample space of all r-colorings of the first n integers, and second and even higher moments for the number of monochromatic complete graphs Kk in Kn. In addition to their considerable independent interest, these formulas would hopefully be instrumental
in improving the extremely weak known lower bounds for the asymptotics of Ramsey number.
PhysicalDescription
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electronic resource
Extent
ix, 135 p. : ill.
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Note
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Ph.D.
Note
Includes abstract
Note
Vita
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Includes bibliographical references
Note
(type = statement of responsibility)
by Thotsaporn "Aek" Thanatipanonda
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Thanatipanonda
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Thotsaporn
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Thotsaporn Thanatipanonda
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Zeilberger
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Doron
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chair
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Doron Zeilberger
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Retakh
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Vladimir
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Vladimir Retakh
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Saks
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Michael
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Michael Saks
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Sloane
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Neil
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Neil Sloane
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Rutgers University
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Graduate School - New Brunswick
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2008
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2008
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xx
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Rutgers University Electronic Theses and Dissertations
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ETD
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Title
Graduate School - New Brunswick Electronic Theses and Dissertations
Identifier
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rucore19991600001
Identifier
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doi:10.7282/T30G3K3F
Genre
(authority = ExL-Esploro)
ETD doctoral
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Rights
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The author owns the copyright to this work.
Copyright
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Copyright protected
Availability
Status
Open
Reason
Permission or license
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Name
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Thanatipanonda
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Thotsaporn
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DateTime
2008-09-24 17:22:35
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Thotsaporn Thanatipanonda
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Rutgers University. Graduate School - New Brunswick
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Author Agreement License
Detail
I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
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