Diagonal Forms, Linear Algebraic Methods and Ramsey-Type Problems

This thesis focuses mainly on linear algebraic aspects of combinatorics. Let Nt(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 Nt(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 zerosum 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 Ramseytype problem on equipartitions, Hartman’s conjecture on large sets of designs and a matroid theory problem proposed by Welsh.