Diagonal Forms, Linear Algebraic Methods and Ramsey-Type Problems

• Main Author: Wong, Wing Hong Tony
• Format: Others
• Published: 2013
• Online Access: https://thesis.library.caltech.edu/7801/1/Thesis.pdf; 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. https://resolver.caltech.edu/CaltechTHESIS:05312013-153531964 <https://resolver.caltech.edu/CaltechTHESIS:05312013-153531964>

<p>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.</p> <p>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.</p> <p>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.</p> <p>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.</p>