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Computational advances in Rado numbers

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Title
Computational advances in Rado numbers
Name (type = personal)
NamePart (type = family)
Myers
NamePart (type = given)
Kellen John
NamePart (type = date)
1985-
DisplayForm
Kellen John Myers
Role
RoleTerm (authority = RULIB)
author
Name (type = personal)
NamePart (type = family)
Zeilberger
NamePart (type = given)
Doron
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Doron Zeilberger
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Advisory Committee
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chair
Name (type = personal)
NamePart (type = family)
Roberts
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Fred
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Fred Roberts
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Advisory Committee
Role
RoleTerm (authority = RULIB)
internal member
Name (type = personal)
NamePart (type = family)
Sloane
NamePart (type = given)
Neil
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Neil Sloane
Affiliation
Advisory Committee
Role
RoleTerm (authority = RULIB)
internal member
Name (type = personal)
NamePart (type = family)
Schaal
NamePart (type = given)
Daniel
DisplayForm
Daniel Schaal
Affiliation
Advisory Committee
Role
RoleTerm (authority = RULIB)
outside member
Name (type = corporate)
NamePart
Rutgers University
Role
RoleTerm (authority = RULIB)
degree grantor
Name (type = corporate)
NamePart
Graduate School - New Brunswick
Role
RoleTerm (authority = RULIB)
school
TypeOfResource
Text
Genre (authority = marcgt)
theses
OriginInfo
DateCreated (qualifier = exact)
2015
DateOther (qualifier = exact); (type = degree)
2015-05
CopyrightDate (encoding = w3cdtf); (qualifier = exact)
2015
Place
PlaceTerm (type = code)
xx
Language
LanguageTerm (authority = ISO639-2b); (type = code)
eng
Abstract (type = abstract)
In this dissertation, we present new methods in the computation of Rado numbers. These methods are applied to several families of equations. The Rado number of an equation is a Ramsey-theoretic quantity associated to the equation. For any particular equation E, the Rado number R_r(E) is the smallest N such that any r-coloring chi:{1,2,...,N} -> {1,2,...,r} must induce a monochromatic solution to E. We will lay out the history of this field and provide some structure as context for new results. Then we will discuss the new methods and computational tools that provide the foundation of the thesis. The 2-color Rado numbers R_2(2x+2y+kz = 3w) and R_2(kx+(k+1)y = (k+2)z) are computed for small values of the parameter k. The 2-color off-diagonal Rado numbers R_2(x + ay = z; x + by = z) are provided for 1 <= a, b <= 20. Likewise, 3-color off-diagonal Rado numbers R_2(x + y = az; x + y = bz; x + y = cz) are computed for 1 <= a, b, c <= 5. We confirm the long-standing conjecture that the 3-color generalized Schur numbers R_3(x_1 + x_2 + ... + x_{m-1} = xm ) are m^3 - m^2 - m - 1 for m = 7, 8, 9, 10 (effectively doubling the empirical evidence for the conjecture) and provide the related Rado numbers R_3( x1 + ... + x(m-2) + k x(m-1) = xm ) for certain (k,m) values. We prove a lower bound for the r-color non-homogeneous Schur numbers: R_r( x + y + c = z) >= (3^r - 1)(c+1)/2 for c >= 0. We also compute the precise values for r = 4 and -20 <= c <= 7 and generalize this bound for m >= 3 variables. We provide the 2-color Rado numbers for 1/x + 1/y = 1/z and a few other equations involving reciprocals. We also construct a coloring proving R_2(x^2 + y^2 = z^2) > 6500. (It is not known whether this Rado number is finite.) We compute the 2- and 3-color Rado numbers for other sums-of-squares equations, sum_{i=1}^a x_i^2) = sum_{i=1}^b y_i^2, and we prove a universal upper bound for a <= b <= ca for a constant c between 1 and 2 (different values of c give different upper bounds). We follow this with Rado numbers for other assorted families of quadratic equations. We also present quantitative analogues of Hindman's theorem, which guarantees monochromatic solutions to systems like {x+y+z = w; x*y*z = v}. We conclude by suggesting a number of conjectures, extensions, and generalizations of these results for future work.
Subject (authority = RUETD)
Topic
Mathematics
Subject (authority = ETD-LCSH)
Topic
Ramsey theory
RelatedItem (type = host)
TitleInfo
Title
Rutgers University Electronic Theses and Dissertations
Identifier (type = RULIB)
ETD
Identifier
ETD_6342
PhysicalDescription
Form (authority = gmd)
electronic resource
InternetMediaType
application/pdf
InternetMediaType
text/xml
Extent
1 online resource (ix, 124 p. : ill.)
Note (type = degree)
Ph.D.
Note (type = bibliography)
Includes bibliographical references
Note (type = statement of responsibility)
by Kellen John Myers
RelatedItem (type = host)
TitleInfo
Title
Graduate School - New Brunswick Electronic Theses and Dissertations
Identifier (type = local)
rucore19991600001
Location
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NjNbRU
Identifier (type = doi)
doi:10.7282/T3GH9KTT
Genre (authority = ExL-Esploro)
ETD doctoral
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RightsDeclaration (ID = rulibRdec0006)
The author owns the copyright to this work.
RightsHolder (type = personal)
Name
FamilyName
Myers
GivenName
Kellen
MiddleName
John
Role
Copyright Holder
RightsEvent
Type
Permission or license
DateTime (encoding = w3cdtf); (qualifier = exact); (point = start)
2015-04-13 16:05:46
AssociatedEntity
Name
Kellen Myers
Role
Copyright holder
Affiliation
Rutgers University. Graduate School - New Brunswick
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License
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Author Agreement License
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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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Copyright protected
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Status
Open
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ETD
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