Doctor of Philosophy
This dissertation is divided into two major sections. Chapters 1 to 4 are concerned with Turán type problems for disconnected graphs and hypergraphs. In Chapter 5, we discuss an unrelated problem dealing with the equivalence of two notions of stationary processes. The Turán number of a graph H, ex(n,H), is the maximum number of edges in any n-vertex graph which is H-free. We discuss the history and results in this area, focusing particularly on the degenerate case for bipartite graphs. Let Pl denote a path on l vertices, and k*Pl denote k vertex-disjoint copies of Pl. We determine ex(n,k*P3) for n appropriately large, confirming a conjecture of Gorgol. Further, we determine ex(n,k*Pl) for arbitrary l, and n appropriately large. We provide background on the famous Erdös-Sós conjecture, and conditional on its truth we determine ex(n,H) when H is an equibipartite forest, for appropriately large n. In Chapter 4, we prove similar results in hypergraphs. We first discuss the related results for extremal numbers of hyperpaths, before proving the extremal numbers for multiple copies of a loose path of fixed length, and the corresponding result for linear paths. We extend this result to forests of loose hyperpaths, and linear hyperpaths. We note here that our results for loose paths, while tight, do not give the extremal numbers in their classical form; much more detail on this is given in Chapter 4. InChapter 5, we discuss two notions of stationary processes. Roughly, a process is a uniform martingale if it can be approximated arbitrarily well by a process in which the letter distribution depends only on a finite amount of the past. A random Markov process is a process with a coupled `look back' time; that is, to determine the letter distribution, it suffices to choose a random look-back time, and then the distribution depends only on the past up to this time. Kalikow proved that on a binary alphabet, any uniform martingale is also a random Markov process. We extend this result to any finite alphabet.
Dissertation or thesis originally submitted to the local University of Memphis Electronic Theses & dissertation (ETD) Repository.
Bushaw, Neal Owen, "Problems in Extremal Combinatorics" (2012). Electronic Theses and Dissertations. 417.