Results in Extremal Graph and Hypergraph Theory

In graph theory, as in many fields of mathematics, one is often interested in finding the maxima or minima of certain functions and identifying the points of optimality. We consider a variety of functions on graphs and hypegraphs and determine the structures that optimize them. A central problem in extremal (hyper)graph theory is that of finding the maximum number of edges in a (hyper)graph that does not contain a specified forbidden substructure. Given an integer n, we consider hypergraphs on n vertices that do not contain a strong simplex, a structure closely related to and containing a simplex. We determine that, for n sufficiently large, the number of edges is maximized by a star. We denote by F (G, r, k) the number of edge r-colorings of a graph G that do not contain a monochromatic clique of size k. Given an integer n, we consider the problem of maximizing this function over all graphs on n vertices. We determine that, for large n, the optimal structures are (k − 1)-partite Turán graphs when r = 4 and k ∈ {3, 4} are fixed. We call a graph F color-critical if it contains an edge whose deletion reduces the chromatic number of F and denote by F (H) the number of copies of the specified color-critical graph F that a graph H contains. Given integers n and m, we consider the minimum of F (H) over all graphs H on n vertices and m edges. The Turán number of F , denoted ex(n, F ), is the largest m for which the minimum of F (H) is zero. We determine that the optimal structures are supergraphs of Turán graphs when n is large and ex(n, F ) ≤ m ≤ ex(n, F )+cn for some c > 0.