This thesis considers three related structures on finite sets and outstanding conjectures on two of them. Several new problems and conjectures are stated.A union-closed collection of sets is a collection of sets which contains the union of each pair of sets in the collection. A completely separating system of sets is a collection of sets in which for each pair of elements of the universal set, there exists a set in the collection which contains the first element but not the second, and another set which contains the second element but not the first. An antichain (Sperner Family) is a collection of distinct sets in which no set is a subset of another set in the collection. The size of an antichain is the number of sets in the collection. The volume of an antichain is the sum of the cardinalities of the sets in the collection. A flat antichain is an antichain in which the difference in cardinality between any two sets in the antichain is at most one.The two outstanding conjectures considered are:The union-closed sets conjecture - In any union-closed collection of non-empty sets there is an element of the universal set in at least half of the sets in the collection;The flat antichain conjecture - Given an antichain with size s and volume V, there is a flat antichain with the same size and volume.Union-closed collections are considered in two ways. Improvements are made to the previously known bounds concerning the minimum size of a counterexample to the union-closed sets conjecture. Results are derived on the minimum size of a union-closed collection generated by a given number of k-sets. An ordering on sets is described, called order R and it is conjectured that choosing a collection of m k-sets in order R will always minimise the size of the union-closed collection generated by m k-sets.Several variants on completely separating systems of sets are considered. A ++ / determination is made of the minimum size of such collections, subject to various constraints on the collections. In particular, for each n and k, exact values or bounds are determined for the minimum size of completely separating systems on a n-set in which each set has cardinality k.Antichains are considered in their relationship to completely separating systems and the flat antichain conjecture is shown to be true in certain cases.

Thesis (Ph. D.)--University of Texas at Austin, 1998. / Vita. Includes bibliographical references (leaves 95-96). Available also in a digital version from Dissertation Abstracts.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1984. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 83-84).

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Thesis (Ph. D.)--Rutgers University, 2009. / "Graduate Program in Mathematics." Includes bibliographical references (p. 57-58).

No description available.

Thesis (Ph.D) - Swinburne University of Technology, Faculty of Information & Communication Technologies, 2008. / [A thesis submitted in total fulfillment of the requirements of for the degree of Doctor of Philosophy, Faculty of Information and Communication Technologies, Swinburne University of Technology, 2008]. Typescript. Includes bibliographical references p. 193-201.

Diplomarbeit--Rheinische Friedrich-Wilhelms-Universität, 2003. / Includes bibliographical references (p. 102-103).

The focus of this thesis is extremal dependence among spatial observations. In particular, this research extends the notion of the extremogram to the spatial process setting. Proposed by Davis and Mikosch (2009), the extremogram measures extremal dependence for a stationary time series. The versatility and flexibility of the concept made it well suited for many time series applications including from finance and environmental science. After defining the spatial extremogram, we investigate the asymptotic properties of the empirical estimator of the spatial extremogram. To this end, two sampling scenarios are considered: 1) observations are taken on the lattice and 2) observations are taken on a continuous region in a continuous space, in which the locations are points of a homogeneous Poisson point process. For both cases, we establish the central limit theorem for the empirical spatial extremogram under general mixing and dependence conditions. A high level overview is as follows. When observations are observed on a lattice, the asymptotic results generalize those obtained in Davis and Mikosch (2009). For non-lattice cases, we define a kernel estimator of the empirical spatial extremogram and establish the central limit theorem provided the bandwidth of the kernel gets smaller and the sampling region grows at proper speeds. We illustrate the performance of the empirical spatial extremogram using simulation examples, and then demonstrate the practical use of our results with a data set of rainfall in Florida and ground-level ozone data in the eastern United States. The second part of the thesis is devoted to bootstrapping and variance estimation with a view towards constructing asymptotically correct confidence intervals. Even though the empirical spatial extremogram is asymptotically normal, the limiting variance is intractable. We consider three approaches: for lattice data, we use the circular bootstrap adapted to spatial observations, jackknife variance estimation, and subsampling variance estimation. For data sampled according to a Poisson process, we use subsampling methods to estimate the variance of the empirical spatial extremogram. We establish the (conditional) asymptotic normality for the circular block bootstrap estimator for the spatial extremogram and show L2 consistency of the variance estimated by jackknife and subsampling. Then, we propose a portmanteau style test to check the existence of extremal dependences at multiple lags. The validity of confidence intervals produced from these approaches and a portmanteau style test are demonstrated through simulation examples. Finally, we illustrate this methodology to two data sets. The first is the amount of rainfall over a grid of locations in northern Florida. The second is ground-level ozone in the eastern United States, which are recorded on an irregularly spaced set of stations.

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