Mantel's Theorem for random graphs

For a graph $G$, denote by $t(G)$ (resp. $b(G)$) the maximum size of a triangle-free (resp. bipartite) subgraph of $G$. Of course $t(G) \geq b(G)$ for any $G$, and a classic result of Mantel from 1907 (the first case of Tur\'an's Theorem) says that equality holds for complete graphs. A natural question, first considered by Babai, Simonovits and Spencer about 20 years ago is, when (i.e. for what $p=p(n)$) is the "Erd\H{o}s-R\'enyi" random graph $G=G(n,p)$ likely to satisfy $t(G) = b(G)$? We show that this is true if $p>C n^{-1/2} \log^{1/2}n$ for a suitable constant $C$, which is best possible up to the value of $C$.Comment: 15 page

Turán problems in graphs and hypergraphs

Mantel's theorem says that among all triangle-free graphs of a given order the balanced complete bipartite graph is the unique graph of maximum size. In Chapter 2, we prove an analogue of this result for 3-graphs (3-uniform hy¬pergraphs) together with an associated stability result. Let K− 4 , F5 and F6 be 3-graphs with vertex sets {1, 2,3, 4}, {1, 2,3,4, 5} and {1, 2,3,4, 5, 6} re¬spectively and edge sets E(K−4 ) = {123, 124, 134}, E(F5) = {123, 124, 345}, E(F6) = {123, 124,345, 156} and F = {K4, F6}. For n =6 5 the unique F-free 3-graph of order n and maximum size is the balanced complete tri¬partite 3-graph S3(n). This extends an old result of Bollobas that S3(n) is the unique 3-graph of maximum size with no copy of K− 4 or F5. In 1941, Turán generalised Mantel's theorem to cliques of arbitrary size and then asked whether similar results could be obtained for cliques on hyper-graphs. This has become one of the central unsolved problems in the field of extremal combinatorics. In Chapter 3, we prove that the Turán density of K(3) 5 together with six other induced subgraphs is 3/4. This is analogous to a similar result obtained for K(3) 4 by Razborov. In Chapter 4, we consider various generalisations of the Turán density. For example, we prove that, if the density in C of ¯P3 is x and C is K3-free, then |E(C)| /(n ) ≤ 1/4+(1/4)J1 − (8/3)x. This is motivated by the observation 2 that the extremal graph for K3 is ¯P3-free, so that the upper bound is a natural extension of a stability result for K3. The question how many edges can be deleted from a blow-up of H before it is H-free subject to the constraint that the same proportion of edges are deleted from each connected pair of vertex sets has become known as the Turán density problem. In Chapter 5, using entropy compression supplemented with some analytic methods, we derive an upper bound of 1 − 1/('y(Δ(H) − /3)), where Δ(H) is the maximum degree of H, 3 ≤ 'y < 4 and /3 ≤ 1. The new bound asymptotically approaches the existing best upper bound despite being derived in a completely different way. The techniques used in these results, illustrating their breadth and connec¬tions between them, are set out in Chapter 1

Graphs where every k-subset of vertices is an identifying set

Let $G=(V,E)$ be an undirected graph without loops and multiple edges. A subset $C\subseteq V$ is called \emph{identifying} if for every vertex $x\in V$ the intersection of $C$ and the closed neighbourhood of $x$ is nonempty, and these intersections are different for different vertices $x$. Let $k$ be a positive integer. We will consider graphs where \emph{every} $k$-subset is identifying. We prove that for every $k>1$ the maximal order of such a graph is at most $2k-2.$ Constructions attaining the maximal order are given for infinitely many values of $k.$ The corresponding problem of $k$-subsets identifying any at most $\ell$ vertices is considered as well.Comment: 21 page

Extremal problems on counting combinatorial structures

The fast developing field of extremal combinatorics provides a diverse spectrum of powerful tools with many applications to economics, computer science, and optimization theory. In this thesis, we focus on counting and coloring problems in this field. The complete balanced bipartite graph on $n$ vertices has \floor{n^2/4} edges. Since all of its subgraphs are triangle-free, the number of (labeled) triangle-free graphs on $n$ vertices is at least 2^{\floor{n^2/4}}. This was shown to be the correct order of magnitude in a celebrated paper Erd\H{o}s, Kleitman, and Rothschild from 1976, where the authors furthermore proved that almost all triangle-free graphs are bipartite. In Chapters 2 and 3 we study analogous problems for triangle-free graphs that are maximal with respect to inclusion. In Chapter 2, we solve the following problem of Paul Erd\H{o}s: Determine or estimate the number of maximal triangle-free graphs on $n$ vertices. We show that the number of maximal triangle-free graphs is at most $2^{n^2/8+o(n^2)}$, which matches the previously known lower bound. Our proof uses among other tools the Ruzsa-Szemer\'{e}di Triangle Removal Lemma and recent results on characterizing of the structure of independent sets in hypergraphs. This is a joint work with J\'{o}zsef Balogh. In Chapter 3, we investigate the structure of maximal triangle-free graphs. We prove that almost all maximal triangle-free graphs admit a vertex partition $(X, Y)$ such that $G[X]$ is a perfect matching and $Y$ is an independent set. Our proof uses the Ruzsa-Szemer\'{e}di Removal Lemma, the Erd\H{o}s-Simonovits stability theorem, and recent results of Balogh-Morris-Samotij and Saxton-Thomason on the characterization of the structure of independent sets in hypergraphs. The proof also relies on a new bound on the number of maximal independent sets in triangle-free graphs with many vertex-disjoint $P_3$'s, which is of independent interest. This is a joint work with J\'{o}zsef Balogh, Hong Liu, and Maryam Sharifzadeh. In Chapte 4, we seek families in posets with the smallest number of comparable pairs. Given a poset $P$, a family \F\subseteq P is \emph{centered} if it is obtained by `taking sets as close to the middle layer as possible'. A poset $P$ is said to have the \emph{centeredness property} if for any $M$, among all families of size $M$ in $P$, centered families contain the minimum number of comparable pairs. Kleitman showed that the Boolean lattice $\{0,1\}^n$ has the centeredness property. It was conjectured by Noel, Scott, and Sudakov, and by Balogh and Wagner, that the poset $\{0,1,\ldots,k\}^n$ also has the centeredness property, provided $n$ is sufficiently large compared to $k$. We show that this conjecture is false for all $k\geq 2$ and investigate the range of $M$ for which it holds. Further, we improve a result of Noel, Scott, and Sudakov by showing that the poset of subspaces of $\mathbb{F}_q^n$ has the centeredness property. Several open problems are also given. This is a joint result with J\'{o}zsef Balogh and Adam Zsolt Wagner. In Chapter 5, we consider a graph coloring problem. Kim and Park have found an infinite family of graphs whose squares are not chromatic-choosable. Xuding Zhu asked whether there is some $k$ such that all $k$-th power graphs are chromatic-choosable. We answer this question in the negative: we show that there is a positive constant $c$ such that for any $k$ there is a family of graphs $G$ with $\chi(G^k)$ unbounded and $\chi_{\ell}(G^k)\geq c \chi(G^k) \log \chi(G^k)$. We also provide an upper bound, $\chi_{\ell}(G^k)1$. This is a joint work with Nicholas Kosar, Benjamin Reiniger, and Elyse Yeager

Labour supply and the "law of demand".

The well-known "law of supply and demand" says that an increase in the price of a commodity leads to a decrease in the aggregate demand for this commodity and an increase in aggregate supply. There is, however, no theoretical foundation for this "law". Empirical evidence, on the other hand, should be interpreted with care. If one estimates the parameters of certain functional forms for demand and supply functions, then the results may simply be consequences of the parametric assumptions made in estimation. The first chapter of the thesis discusses the implications of the assumption of profit and utility maximisation for the properties of demand and supply functions. It explains why economic rationality on the microlevel does not, in general, lead to macroeconomic regularities and suggests replacing the consumption sector of the neoclassical equilibrium model by a large population of individually small consumers. Such a population will be explored in the second chapter. The chapter is a direct outgrowth of a basic contribution by W. Hildenbrand: "On the Law of Demand", Econometrica 1983. In W. Hildenbrand's model the market demand function is defined by integrating an individual demand function with respect to an exogenously given income distribution. We build into the model an individual labour supply function and then compare the matrix of aggregate income effects studied by W. Hildenbrand with that obtained by integrating the individual demand function with respect to a distribution of wage rates. The empirical part of the thesis analyses the labour supply and earnings data in the U.K. Family Expenditure Survey 1970-85. Using non- parametric smoothing methods, the elasticity of labour supply with respect to the wage rate is estimated for several groups of workers. The estimations for full-time workers confirm the famous "downward sloping" labour supply function. The estimated elasticities for the entire population of workers for the years 1970-85 have the mean value 0.2 and the standard deviation 0.02

Decision making theory with geographic information systems support

Decisions are made with varying degrees of effectiveness and efficiency and are influenced by a myriad of internal and external forces. Decision Support Systems (DSS) software can effectively aid decision making through processing the facts and producing meaningful outputs for use by the person or team in making the final choice. Geographic Information Systems (GIS), a form of DSS, are very effective when locational data are present. This thesis talks about using GIS software in decision making procedures

A cognitive model of fiction writing.

Models of the writing process are used to design software tools for writers who work with computers. This thesis is concerned with the construction of a model of fiction writing. The first stage in this construction is to review existing models of writing. Models of writing used in software design and writing research include behavioural, cognitive and linguistic varieties. The arguments of this thesis are, firstly, that current models do not provide an adequate basis for designing software tools for fiction writers. Secondly, research into writing is often based on questionable assumptions concerning language and linguistics, the interpretation of empirical research, and the development of cognitive models. It is argued that Saussure's linguistics provides an alternative basis for developing a model of fiction writing, and that Barthes' method of textual analysis provides insight into the ways in which readers and writers create meanings. The result of reviewing current models of writing is a basic model of writing, consisting of a cycle of three activities - thinking, writing, and reading. The next stage is to develop this basic model into a model of fiction writing by using narratology, textual analysis, and cognitive psychology to identify the kinds of thinking processes that create fictional texts. Remembering and imagining events and scenes are identified as basic processes in fiction writing; in cognitive terms, events are verbal representations, while scenes are visual representations. Syntax is identified as another distinct object of thought, to which the processes of remembering and imagining also apply. Genette's notion of focus in his analysis of text types is used to describe the role of characters in the writer's imagination: focusing the imagination is a process in which a writer imagines she is someone else, and it is shown how this process applies to events, scenes, and syntax. It is argued that a writer's story memory, influences his remembering and imagining; Todorov's work on symbolism is used to argue that interpretation plays the role in fiction writing of binding together these two processes. The role of naming in reading and its relation to problem solving is compared with its role in writing, and names or signifiers are added to the objects of thought in fiction writing. It is argued that problem solving in fiction writing is sometimes concerned with creating problems or mysteries for the reader, and it is shown how this process applies to events, scenes, signifiers and syntax. All these findings are presented in the form of a cognitive model of fiction writing. The question of testing is discussed, and the use of the model in designing software tools is illustrated by the description of a hypertextual aid for fiction writers

Investigating tuberculosis transmission using spatial methods

BACKGROUND: Tuberculosis remains a leading infectious cause of death worldwide. Reducing transmission requires an increased focus on local control measures informed by spatial data. Effective use of spatial methods will improve understanding of tuberculosis transmission and support outbreak investigations. METHODS: I conducted a systematic literature review to describe spatial methods that have been used in previous outbreak investigations (Chapter 2). I developed and evaluated a novel interactive mapping tool, written using the R programming language (Chapter 3). Using multinomial logistic regression and spatial scan statistics, I investigated molecular and spatial clustering of tuberculosis in London (Chapter 4). I described the evolution of a large outbreak of drug-resistant tuberculosis in London in space and time (Chapter 5). Through three case studies, I assessed the utility of a novel spatial tool, geographic profiling, which aims to identify the locations of sources of infectious disease using locations of linked cases (Chapter 6). I analysed the spatial accessibility of tuberculosis services in London using travel time data (Chapter 7). KEY FINDINGS: • Spatial methods provide an important complementary tool to epidemiological analyses, but are currently under-used (less than half a percent of published outbreak investigations used spatial methods). • Large numbers of tuberculosis cases in London have resulted from local transmission, with more than one in ten cases part of large clusters. • Social complexity and area-level deprivation are associated with transmission of tuberculosis in large clusters. • Geographic profiling may assist with epidemiological investigations of infectious diseases in some circumstances by prioritising areas for investigation. • Pan-London commissioning could improve tuberculosis services by enhancing spatial accessibility. CONCLUSIONS: Spatial methods provide many valuable contributions to investigations of tuberculosis. Development of new tools and wider use of existing methods could limit the public health impacts of infectious disease outbreaks

The life and work of the Revd. T.H.E.C. Espin perpetual curate of tow law with special reference to his astronomical research

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