Citation
Grynkiewicz, David Joseph (2006) Sumsets, zerosums and extremal combinatorics. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd08102005190712
Abstract
This thesis develops and applies a method of tackling zerosum additive questions, especially those related to the ErdosGinzburgZiv Theorem (EGZ), through the use of partitioning sequences into sets, i.e., set partitions. Much of the research can alternatively be found in the literature spread across nine separate articles, but is here collected into one cohesive work augmented by additional exposition. Highlights include a new combinatorial proof of Kneser's Theorem (not currently located elsewhere); a proof of Caro's conjectured weighted ErdosGinzburgZiv Theorem; a partition analog of the CauchyDavenport Theorem that encompasses several results of Mann, Olson, Bollobas and Leader, and Hamidoune; a refinement of EGZ showing that an essentially dichromatic sequence of 2m1 terms from an abelian group of order m contains a mostly monochromatic mterm zerosum subsequence; an interpretation of Kemperman's Structure Theorem (KST) for critical pairs (i.e., those finite subsets A and B of an abelian group with A+B<A+B) through quasiperiodic decompositions, which establishes certain canonical aspects of KST and facilitates its use in practice; a draining theorem for set partitions showing that a set partition of large cardinality sumset can have several elements removed from its terms and still have the sumset remain of large cardinality; a proof of a subsequence sum conjecture of Hamidoune; the determination of the g(m,k) function introduced by Bialostocki and Lotspeich (defined as the least n so that a sequence of terms from Z/mZ of length n with at least k distinct terms must contain an mterm zerosum subsequence) for m large with respect to k; the determination of g(m,5) for all m, including the details to the abbreviated proof found in the literature; various zerosum results concerning modifications to the nondecreasing diameter problem of Bialostocki, Erdos, and Lefmann; an extension of EGZ to a class of hypergraphs; and a lower bound on the number of zerosum mterm subsequences in a sequence of n terms from an abelian group of order m that establishes Bialostocki's conjectured value for small n<(19/3)m .
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  additive number theory; ErdosGinzburgZiv; extremal combinatorics; set partition; sumset; zerosum 
Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  10 August 2005 
NonCaltech Author Email:  diambri (AT) hotmail.com 
Record Number:  CaltechETD:etd08102005190712 
Persistent URL:  http://resolver.caltech.edu/CaltechETD:etd08102005190712 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  3085 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  11 Aug 2005 
Last Modified:  26 Dec 2012 02:56 
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