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Diagonal Forms, Linear Algebraic Methods and Ramsey-Type Problems


Wong, Wing Hong Tony (2013) Diagonal Forms, Linear Algebraic Methods and Ramsey-Type Problems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/5B5A-Q252.


This thesis focuses mainly on linear algebraic aspects of combinatorics. Let N_t(H) be an incidence matrix with edges versus all subhypergraphs of a complete hypergraph that are isomorphic to H. Richard M. Wilson and the author find the general formula for the Smith normal form or diagonal form of N_t(H) for all simple graphs H and for a very general class of t-uniform hypergraphs H.

As a continuation, the author determines the formula for diagonal forms of integer matrices obtained from other combinatorial structures, including incidence matrices for subgraphs of a complete bipartite graph and inclusion matrices for multisets.

One major application of diagonal forms is in zero-sum Ramsey theory. For instance, Caro's results in zero-sum Ramsey numbers for graphs and Caro and Yuster's results in zero-sum bipartite Ramsey numbers can be reproduced. These results are further generalized to t-uniform hypergraphs. Other applications include signed bipartite graph designs.

Research results on some other problems are also included in this thesis, such as a Ramsey-type problem on equipartitions, Hartman's conjecture on large sets of designs and a matroid theory problem proposed by Welsh.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Combinatorics, Diagonal Forms, Linear Algebraic Methods, Ramsey Theory
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Wilson, Richard M.
Thesis Committee:
  • Wilson, Richard M. (chair)
  • Omar, Mohamed
  • Ramakrishnan, Dinakar
  • Wales, David B.
Defense Date:28 May 2013
Non-Caltech Author Email:tonywhwong (AT)
Funding AgencyGrant Number
Sir Edward Youde Memorial FundUNSPECIFIED
Record Number:CaltechTHESIS:05312013-153531964
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:7801
Deposited By: Wing Hong Tony Wong
Deposited On:03 Jun 2013 22:46
Last Modified:04 Oct 2019 00:01

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