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 continu...

We say that a graph G has the Ramsey property w.r.t. some graph F and some integer r ≥ 2, or G is (F, r)-Ramsey for short, if any r-coloring of the edges of G contains a monochromatic copy of F . Rödl and Ruciński asked how globally sparse (F, r)-Ramsey graphs G can possibly be, where the density of G is measured by the subgraph H ⊆ G with the highest average degree. So far, this so-called Rams...

Let Kl×t be a complete, balanced, multipartite graph consisting of l partite sets and t vertices in each partite set. For given two graphs G1 and G2, and integer j ≥ 2, the size multipartite Ramsey number mj(G1, G2) is the smallest integer t such that every factorization of the graph Kj×t := F1 ⊕ F2 satisfies the following condition: either F1 contains G1 or F2 contains G2. In 2007, Syafrizal e...

Geometric versions of Ramsey-type and Turán-type problems are studied in a special but natural representation of bipartite graphs and similar questions are asked for general representations. A bipartite geometric graphG(m,n) = [A,B] is simple if the vertex classes A,B of G(m,n) are represented in R2 as A = {(1, 0), (2, 0), . . . , (m, 0)}, B = {(1, 1), (2, 1), . . . , (n, 1)} and the edge ab is...