• Login
    View Item 
    •   SMARTech Home
    • Georgia Tech Theses and Dissertations
    • Georgia Tech Theses and Dissertations
    • View Item
    •   SMARTech Home
    • Georgia Tech Theses and Dissertations
    • Georgia Tech Theses and Dissertations
    • View Item
    JavaScript is disabled for your browser. Some features of this site may not work without it.

    New combinatorial techniques for nonlinear orders

    Thumbnail
    View/Open
    marcus_adam_w_200808_phd.pdf (405.7Kb)
    Date
    2008-06-26
    Author
    Marcus, Adam Wade
    Metadata
    Show full item record
    Abstract
    This thesis focuses on the use of extremal techniques in analyzing problems that historically have been associated with other areas of discrete mathematics. We establish new techniques for analyzing combinatorial problems with two different types of nonlinear orders, and then use them to solve important previously-open problems in mathematics. In addition, we use entropy techniques to establish a variety of bounds in the theory of sumsets. In the second chapter, we examine a problem of Furedi and Hajnal regarding forbidden patterns in (0,1)-matrices. We introduce a new technique that gives an asymptotically tight bound on the number of 1-entries that a (0,1)-matrix can contain while avoiding a fixed permutation matrix. We use this result to solve the Stanley-Wilf conjecture, a well-studied open problem in enumerative combinatorics. We then generalize the technique to give results on d-dimensional matrices. In the third chapter, we examine a problem of Pinchasi and Radoicic on cyclically order sets. To do so, we prove an upper bound on the sizes of such sets, given that their orders have the intersection reverse property. We then use this to give an upper bound on the number of edges that a graph can have, assuming that the graph can be drawn so that no cycle of length four has intersecting edges. This improves the previously best known bound and (up to a log-factor) matches the best known lower bound. This result implies improved bounds on a number of well-studied problems in geometric combinatorics, most notably the complexity of pseudo-circle arrangements. In the final chapter, we use entropy techniques to establish new bounds in the theory of sumsets. We show that such sets behave fractionally submultiplicatively, which in turn provides new Plunecke-type inequalities of the form first introduced by Gyarmati, Matolcsi, and Ruzsa.
    URI
    http://hdl.handle.net/1853/24685
    Collections
    • Georgia Tech Theses and Dissertations [22401]
    • School of Mathematics Theses and Dissertations [399]

    Browse

    All of SMARTechCommunities & CollectionsDatesAuthorsTitlesSubjectsTypesThis CollectionDatesAuthorsTitlesSubjectsTypes

    My SMARTech

    Login

    Statistics

    View Usage StatisticsView Google Analytics Statistics
    facebook instagram twitter youtube
    • My Account
    • Contact us
    • Directory
    • Campus Map
    • Support/Give
    • Library Accessibility
      • About SMARTech
      • SMARTech Terms of Use
    Georgia Tech Library266 4th Street NW, Atlanta, GA 30332
    404.894.4500
    • Emergency Information
    • Legal and Privacy Information
    • Human Trafficking Notice
    • Accessibility
    • Accountability
    • Accreditation
    • Employment
    © 2020 Georgia Institute of Technology