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Turan Numbers Faculty of Sciences Department of Mathematics Academic year 2015–2016 Finite Geometry and Graph Theory intertwine: Turán Numbers Sam Mattheus Promotor: Prof. dr. L. Storme Master thesis submied to obtain the academic degree of Master in Mathematics, specialization Pure Mathematics. Faculty of Sciences Department of Mathematics Academic year 2015–2016 Finite Geometry and Graph Theory intertwine: Turán Numbers Sam Mattheus Promotor: Prof. dr. L. Storme Master thesis submied to obtain the academic degree of Master in Mathematics, specialization Pure Mathematics. Introduction Extremal graph theory, in its strictest sense, is a branch of graph theory developed and loved by Hungarians. Béla Bollobás Extremal graph theory, and most of combinatorics for that maer, are relatively recent subjects in mathematics, compared to for example analysis or number theory. It has seen a lot of growth during the 20th century, mainly under the impulse of Erdős and other Hungarian mathematicians, as Bollobás also remarked. In this thesis we will discuss a subject in extremal graph theory which originates from a theorem by another Hungarian mathematician, Pál Turán. In 1941 he investigated the following problem: what is the maximal number of edges a graph on n vertices can have, without containing a complete graph Kr? This result was the genesis of a subfield of extremal graph theory which is nowadays called Turán problems. In this thesis, we will focus on the following problem: what is the maximal number of edges a graph on n vertices can have, without containing a fixed graph H? This problem is also known as the forbidden subgraph problem. The maximum is denoted as ex(n; H) and studied as a function of n. When exact values are known of this function for certain n, one then tries to investigate the extremal graphs, which are denoted as EX(n; H). When H is not bipartite, Erdős, Stone and Simonovits determined the asymptotic growth of ex(n; H), this is called the non-degenerate case. In the degenerate case, when H is bipartite, there is not much known. This is the case on which we will focus most. The general procedure to investigate ex(n; H) in the degenerate case is twofold: on the one hand, one tries to find upper bounds, based on graph- theoretical arguments. On the other hand, using for example finite geometries, one tries to find matching lower bounds. This is where the interplay between graph theory and finite geometry will appear most clearly. In Chapter 1 we investigate the function ex(n; H) for arbitrary H and look at a few instances where the values of these functions are known for all n. This chapter serves as an introduction to the field and develops the necessary graph theory. In Chapter 2 we look at the case when H is the quadrilateral C4 = K2;2. For certain values of n, ex(n; C4) and EX(n; C4) have been determined. We will see that finite projective planes play a big role here and appear naturally when considering C4-free graphs. In Chapter 3 we generalise this to general complete bipartite graphs Ks;t. We will see that designs come into play here. It is no coincidence that projective planes are a subclass of designs. Lastly, in Chapter 4, we generalise C4 to even cycles C2k. Again, there exists finite geometries that lead to good constructions of C2k-free graphs. These are the generalized polygons, which also have projective planes as a subclass. What we will see in these three chapters is that finite geometries lend themselves very well to constructing infinite families of graphs satisfying certain properties. The chapters themselves have following structure: first we introduce some theory which will play a big role in the chapter. In our four chapters, we introduce graph theory, projective planes and spaces, designs and generalized polygons respectively. These are mainly wrien for the reader that is not familiar with these subjects. Veterans in the field will probably be able to quickly skim through these parts, briefly recalling some definitions and concepts in the process. Chapter 1 is a bit of an outsider, as this one does not introduce any finite geometry, and serves mainly as an introduction to graph theory. The other chapters first develop some theory from finite geometry. Then we investigate the known i upper bounds, but the most important part is finding good lower bounds. This is where the theory we develop at the beginning comes into play. We will investigate the opportunities and limits of the finite geometry constructions and end each chapter with some open questions. We will not always mention this explicitly, but every object we work with is finite. Moreover, we make following conventions. The notation N is used for the non-negative integers. Unless explicity stated otherwise, arabic leers will denote non-negative integers, i.e., when we write k or k ≥ 1, we assume k 2 N. On the other hand c > 0 denotes a non-negative real number and > 0 is the usual notation for a small non-negative real number. A set of only one element will oen be wrien without A accolades. If A is a set, then jAj denotes its size and k all subsets of A having size k. If we say ‘take k vertices’, we mean we take k distinct vertices. In the same vein, x; y 2 A will imply that x 6= y. Lastly, we use following asymptotic notation. Let f(n) and g(n) be functions from N to N, then we have following asymptotic notation. • f(n) = O(g(n)) if and only if f(n) ≤ cg(n), where c > 0, for n suiciently large, • f(n) = o(g(n)) if and only if g(n) 6= 0 and limn!1 f(n)=g(n) = 0, • f(n) = Θ(g(n)) if and only if f(n) = O(g(n)) and g(n) = O(f(n)), • f(n) ≈ g(n) if and only if limn!1 f(n)=g(n) = 1, • f(n) = Ω(g(n)) if and only if there are infinitely many n such that cg(n) ≤ f(n) for some c > 0. Using this notation, we can write results very compact. On the other hand, it is not very useful for proofs. If we have to proof a statement containing these notations, we will resort to standard arguments containing , δ; n0, making computations easier. As every other author, we have to make decisions what to include and what to omit. Computational- heavy, long or technical proofs are generally not included. We hope that we have found a good balance between the two. I thank Leo Storme, my promotor, for suggesting this subject. It has been a real treat to work on. His help, suggestions and comments have been invaluable to this thesis, and without him, it would simply not exist. I also thank Francesco Pavese, who took the time and the patience to explain several concepts from finite geometry. The new results regarding the independence number of ERq are the result of the collaboration with him. I thank him deeply for giving me the opportunity to do so. Many thanks go to Ward Poelmans for sharing the LATEX template of his PhD thesis, it can be found online on latex.ugent.be. Lastly, I thank everyone who has helped in any way in creating this thesis. It has been a long and wonderful road. The author gives permission to make this master dissertation available for consultation and to copy parts of this master dissertation for personal use. In the case of any other use, the limitations of the copyright have to be respected, in particular with regard to the obligation to state expressly the source when quoting results from this master dissertation. Sam Maheus Gent, May 31, 2016 ii Contents Introduction i 1 Forbidding subgraphs 1 1.0 Foundations: graph theory . .1 1.0.1 Definitions and notation . .1 1.0.2 Examples of graphs . .3 1.0.3 More graph theoretic properties . .4 1.1 Genesis . .5 1.2 Some exact Turán numbers . 10 1.2.1 Odd cycles . 10 1.2.2 Even wheels . 12 1.2.3 Intersecting triangles . 15 1.2.4 Trees . 16 2 The quadrilateral 19 2.0 Foundations: projective geometry . 19 2.0.1 Projective planes and projective spaces . 19 2.0.2 Polarities of projective spaces . 22 2.1 History . 26 2.2 The polarity graph . 33 2.2.1 q even ......................................... 34 2.2.2 q odd.......................................... 35 2.2.3 Independence number . 40 3 Bipartite complete graphs 47 3.0 Foundations: design theory . 47 3.0.1 Basics . 47 3.0.2 Symmetric designs . 49 3.1 General results . 50 3.2 Finding good lower bounds . 53 3.2.1 The norm graph . 53 iii CONTENTS 3.2.2 Excluding K3;3 .................................... 54 3.3 Excluding K2;t+1 ....................................... 57 3.4 A link to designs? . 57 3.4.1 Designs and K2,λ+1-free graphs . 57 3.4.2 Designs and strongly regular graphs . 57 3.4.3 Extending the correspondence . 59 3.4.4 Application: K2;3-free graphs from biplanes . 61 3.4.5 Conclusion . 64 4 Even cycles 67 4.0 Generalized polygons . 67 4.1 Upper bounds and a probabilistic construction . 71 4.2 The polarity graph for generalized polygons . 72 4.3 Beer C2k-free graphs . 74 A Samenvaing 77 B New construction for Z(m; n; 2; 3) 79 Bibliography 81 iv Chapter 1 Forbidding subgraphs In this chapter we will give an introduction to the earliest and most fundamental results of this branch of extremal graph theory.
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