The Unity of Combinatorics
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Forbidding Subgraphs
Graph Theory and Additive Combinatorics Lecturer: Prof. Yufei Zhao 2 Forbidding subgraphs 2.1 Mantel’s theorem: forbidding a triangle We begin our discussion of extremal graph theory with the following basic question. Question 2.1. What is the maximum number of edges in an n-vertex graph that does not contain a triangle? Bipartite graphs are always triangle-free. A complete bipartite graph, where the vertex set is split equally into two parts (or differing by one vertex, in case n is odd), has n2/4 edges. Mantel’s theorem states that we cannot obtain a better bound: Theorem 2.2 (Mantel). Every triangle-free graph on n vertices has at W. Mantel, "Problem 28 (Solution by H. most bn2/4c edges. Gouwentak, W. Mantel, J. Teixeira de Mattes, F. Schuh and W. A. Wythoff). Wiskundige Opgaven 10, 60 —61, 1907. We will give two proofs of Theorem 2.2. Proof 1. G = (V E) n m Let , a triangle-free graph with vertices and x edges. Observe that for distinct x, y 2 V such that xy 2 E, x and y N(x) must not share neighbors by triangle-freeness. Therefore, d(x) + d(y) ≤ n, which implies that d(x)2 = (d(x) + d(y)) ≤ mn. ∑ ∑ N(y) x2V xy2E y On the other hand, by the handshake lemma, ∑x2V d(x) = 2m. Now by the Cauchy–Schwarz inequality and the equation above, Adjacent vertices have disjoint neigh- borhoods in a triangle-free graph. !2 ! 4m2 = ∑ d(x) ≤ n ∑ d(x)2 ≤ mn2; x2V x2V hence m ≤ n2/4. -
The History of Degenerate (Bipartite) Extremal Graph Problems
The history of degenerate (bipartite) extremal graph problems Zolt´an F¨uredi and Mikl´os Simonovits May 15, 2013 Alfr´ed R´enyi Institute of Mathematics, Budapest, Hungary [email protected] and [email protected] Abstract This paper is a survey on Extremal Graph Theory, primarily fo- cusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions. 1 Contents 1 Introduction 4 1.1 Some central theorems of the field . 5 1.2 Thestructureofthispaper . 6 1.3 Extremalproblems ........................ 8 1.4 Other types of extremal graph problems . 10 1.5 Historicalremarks . .. .. .. .. .. .. 11 arXiv:1306.5167v2 [math.CO] 29 Jun 2013 2 The general theory, classification 12 2.1 The importance of the Degenerate Case . 14 2.2 The asymmetric case of Excluded Bipartite graphs . 15 2.3 Reductions:Hostgraphs. 16 2.4 Excluding complete bipartite graphs . 17 2.5 Probabilistic lower bound . 18 1 Research supported in part by the Hungarian National Science Foundation OTKA 104343, and by the European Research Council Advanced Investigators Grant 267195 (ZF) and by the Hungarian National Science Foundation OTKA 101536, and by the European Research Council Advanced Investigators Grant 321104. (MS). 1 F¨uredi-Simonovits: Degenerate (bipartite) extremal graph problems 2 2.6 Classification of extremal problems . 21 2.7 General conjectures on bipartite graphs . 23 3 Excluding complete bipartite graphs 24 3.1 Bipartite C4-free graphs and the Zarankiewicz problem . 24 3.2 Finite Geometries and the C4-freegraphs . -
Projective Planarity of Matroids of 3-Nets and Biased Graphs
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 76(2) (2020), Pages 299–338 Projective planarity of matroids of 3-nets and biased graphs Rigoberto Florez´ ∗ Deptartment of Mathematical Sciences, The Citadel Charleston, South Carolina 29409 U.S.A. [email protected] Thomas Zaslavsky† Department of Mathematical Sciences, Binghamton University Binghamton, New York 13902-6000 U.S.A. [email protected] Abstract A biased graph is a graph with a class of selected circles (“cycles”, “cir- cuits”), called “balanced”, such that no theta subgraph contains exactly two balanced circles. A 3-node biased graph is equivalent to an abstract partial 3-net. We work in terms of a special kind of 3-node biased graph called a biased expansion of a triangle. Our results apply to all finite 3-node biased graphs because, as we prove, every such biased graph is a subgraph of a finite biased expansion of a triangle. A biased expansion of a triangle is equivalent to a 3-net, which, in turn, is equivalent to an isostrophe class of quasigroups. A biased graph has two natural matroids, the frame matroid and the lift matroid. A classical question in matroid theory is whether a matroid can be embedded in a projective geometry. There is no known general answer, but for matroids of biased graphs it is possible to give algebraic criteria. Zaslavsky has previously given such criteria for embeddability of biased-graphic matroids in Desarguesian projective spaces; in this paper we establish criteria for the remaining case, that is, embeddability in an arbitrary projective plane that is not necessarily Desarguesian. -
Finite Geometries
Finite Geometries Gy¨orgyKiss June 26th, 2012, Rogla gyk Finite Geometries Chain of length h : x0 I x1 I ::: I xh where xi 2 P [ L: The distance of two elements d(x; y) : length of the shortest chain joining them. Generalized polygons Let S = (P; L; I) be a connected, finite point-line incidence geometry. P and L are two distinct sets, the elements of P are called points, the elements of L are called lines. I ⊂ (P × L) [ (L × P) is a symmetric relation, called incidence. gyk Finite Geometries The distance of two elements d(x; y) : length of the shortest chain joining them. Generalized polygons Let S = (P; L; I) be a connected, finite point-line incidence geometry. P and L are two distinct sets, the elements of P are called points, the elements of L are called lines. I ⊂ (P × L) [ (L × P) is a symmetric relation, called incidence. Chain of length h : x0 I x1 I ::: I xh where xi 2 P [ L: gyk Finite Geometries Generalized polygons Let S = (P; L; I) be a connected, finite point-line incidence geometry. P and L are two distinct sets, the elements of P are called points, the elements of L are called lines. I ⊂ (P × L) [ (L × P) is a symmetric relation, called incidence. Chain of length h : x0 I x1 I ::: I xh where xi 2 P [ L: The distance of two elements d(x; y) : length of the shortest chain joining them. gyk Finite Geometries Generalized polygons Definition Let n > 1 be a positive integer. -
Octonion Multiplication and Heawood's
CONFLUENTES MATHEMATICI Bruno SÉVENNEC Octonion multiplication and Heawood’s map Tome 5, no 2 (2013), p. 71-76. <http://cml.cedram.org/item?id=CML_2013__5_2_71_0> © Les auteurs et Confluentes Mathematici, 2013. Tous droits réservés. L’accès aux articles de la revue « Confluentes Mathematici » (http://cml.cedram.org/), implique l’accord avec les condi- tions générales d’utilisation (http://cml.cedram.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation á fin strictement personnelle du copiste est constitutive d’une infrac- tion pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Confluentes Math. 5, 2 (2013) 71-76 OCTONION MULTIPLICATION AND HEAWOOD’S MAP BRUNO SÉVENNEC Abstract. In this note, the octonion multiplication table is recovered from a regular tesse- lation of the equilateral two timensional torus by seven hexagons, also known as Heawood’s map. Almost any article or book dealing with Cayley-Graves algebra O of octonions (to be recalled shortly) has a picture like the following Figure 0.1 representing the so-called ‘Fano plane’, which will be denoted by Π, together with some cyclic ordering on each of its ‘lines’. The Fano plane is a set of seven points, in which seven three-point subsets called ‘lines’ are specified, such that any two points are contained in a unique line, and any two lines intersect in a unique point, giving a so-called (combinatorial) projective plane [8,7]. -
The Fact of Modern Mathematics: Geometry, Logic, and Concept Formation in Kant and Cassirer
THE FACT OF MODERN MATHEMATICS: GEOMETRY, LOGIC, AND CONCEPT FORMATION IN KANT AND CASSIRER by Jeremy Heis B.A., Michigan State University, 1999 Submitted to the Graduate Faculty of Arts and Sciences in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Pittsburgh 2007 UNIVERSITY OF PITTSBURGH COLLEGE OF ARTS AND SCIENCES This dissertation was presented by Jeremy Heis It was defended on September 5, 2007 and approved by Jeremy Avigad, Associate Professor, Philosophy, Carnegie Mellon University Stephen Engstrom, Associate Professor, Philosophy, University of Pittsburgh Anil Gupta, Distinguished Professor, Philosophy, University of Pittsburgh Kenneth Manders, Associate Professor, Philosophy, University of Pittsburgh Thomas Ricketts, Professor, Philosophy, University of Pittsburgh Dissertation Advisor: Mark Wilson, Professor, Philosophy, University of Pittsburgh ii Copyright © by Jeremy Heis 2007 iii THE FACT OF MODERN MATHEMATICS: GEOMETRY, LOGIC, AND CONCEPT FORMATION IN KANT AND CASSIRER Jeremy Heis, PhD University of Pittsburgh, 2007 It is now commonly accepted that any adequate history of late nineteenth and early twentieth century philosophy—and thus of the origins of analytic philosophy—must take seriously the role of Neo-Kantianism and Kant interpretation in the period. This dissertation is a contribution to our understanding of this interesting but poorly understood stage in the history of philosophy. Kant’s theory of the concepts, postulates, and proofs of geometry was informed by philosophical reflection on diagram-based geometry in the Greek synthetic tradition. However, even before the widespread acceptance of non-Euclidean geometry, the projective revolution in nineteenth century geometry eliminated diagrams from proofs and introduced “ideal” elements that could not be given a straightforward interpretation in empirical space. -
Matroids You Have Known
26 MATHEMATICS MAGAZINE Matroids You Have Known DAVID L. NEEL Seattle University Seattle, Washington 98122 [email protected] NANCY ANN NEUDAUER Pacific University Forest Grove, Oregon 97116 nancy@pacificu.edu Anyone who has worked with matroids has come away with the conviction that matroids are one of the richest and most useful ideas of our day. —Gian Carlo Rota [10] Why matroids? Have you noticed hidden connections between seemingly unrelated mathematical ideas? Strange that finding roots of polynomials can tell us important things about how to solve certain ordinary differential equations, or that computing a determinant would have anything to do with finding solutions to a linear system of equations. But this is one of the charming features of mathematics—that disparate objects share similar traits. Properties like independence appear in many contexts. Do you find independence everywhere you look? In 1933, three Harvard Junior Fellows unified this recurring theme in mathematics by defining a new mathematical object that they dubbed matroid [4]. Matroids are everywhere, if only we knew how to look. What led those junior-fellows to matroids? The same thing that will lead us: Ma- troids arise from shared behaviors of vector spaces and graphs. We explore this natural motivation for the matroid through two examples and consider how properties of in- dependence surface. We first consider the two matroids arising from these examples, and later introduce three more that are probably less familiar. Delving deeper, we can find matroids in arrangements of hyperplanes, configurations of points, and geometric lattices, if your tastes run in that direction. -
The Turan Number of the Fano Plane
Combinatorica (5) (2005) 561–574 COMBINATORICA 25 Bolyai Society – Springer-Verlag THE TURAN´ NUMBER OF THE FANO PLANE PETER KEEVASH, BENNY SUDAKOV* Received September 17, 2002 Let PG2(2) be the Fano plane, i.e., the unique hypergraph with 7 triples on 7 vertices in which every pair of vertices is contained in a unique triple. In this paper we prove that for sufficiently large n, the maximum number of edges in a 3-uniform hypergraph on n vertices not containing a Fano plane is n n/2 n/2 ex n, P G (2) = − − . 2 3 3 3 Moreover, the only extremal configuration can be obtained by partitioning an n-element set into two almost equal parts, and taking all the triples that intersect both of them. This extends an earlier result of de Caen and F¨uredi, and proves an old conjecture of V. S´os. In addition, we also prove a stability result for the Fano plane, which says that a 3-uniform hypergraph with density close to 3/4 and no Fano plane is approximately 2-colorable. 1. Introduction Given an r-uniform hypergraph F,theTur´an number of F is the maximum number of edges in an r-uniform hypergraph on n vertices that does not contain a copy of F. We denote this number by ex(n,F). Determining these numbers is one of the central problems in Extremal Combinatorics, and it is well understood for ordinary graphs (the case r =2). It is completely solved for many instances, including all complete graphs. Moreover, asymptotic results are known for all non-bipartite graphs. -
Finite Projective Geometry 2Nd Year Group Project
Finite Projective Geometry 2nd year group project. B. Doyle, B. Voce, W.C Lim, C.H Lo Mathematics Department - Imperial College London Supervisor: Ambrus Pal´ June 7, 2015 Abstract The Fano plane has a strong claim on being the simplest symmetrical object with inbuilt mathematical structure in the universe. This is due to the fact that it is the smallest possible projective plane; a set of points with a subsets of lines satisfying just three axioms. We will begin by developing some theory direct from the axioms and uncovering some of the hidden (and not so hidden) symmetries of the Fano plane. Alternatively, some projective planes can be derived from vector space theory and we shall also explore this and the associated linear maps on these spaces. Finally, with the help of some theory of quadratic forms we will give a proof of the surprising Bruck-Ryser theorem, which shows that if a projective plane has order n congruent to 1 or 2 mod 4, then n is the sum of two squares. Thus we will have demonstrated fascinating links between pure mathematical disciplines by incorporating the use of linear algebra, group the- ory and number theory to explain the geometric world of projective planes. 1 Contents 1 Introduction 3 2 Basic Defintions and results 4 3 The Fano Plane 7 3.1 Isomorphism and Automorphism . 8 3.2 Ovals . 10 4 Projective Geometry with fields 12 4.1 Constructing Projective Planes from fields . 12 4.2 Order of Projective Planes over fields . 14 5 Bruck-Ryser 17 A Appendix - Rings and Fields 22 2 1 Introduction Projective planes are geometrical objects that consist of a set of elements called points and sub- sets of these elements called lines constructed following three basic axioms which give the re- sulting object a remarkable level of symmetry. -
Geometry of the Mathieu Groups and Golay Codes
Proc. Indian Acad. Sci. (Math. Sci.), Vol. 98, Nos 2 and 3, December 1988, pp. 153-177. 9 Printed in India. Geometry of the Mathieu groups and Golay codes ERIC A LORD Centre for Theoretical Studies and Department of Applied Mathematics, Indian Institute of Science, Bangalore 560012, India MS received 11 January 1988 Abstraet. A brief review is given of the linear fractional subgroups of the Mathieu groups. The main part of the paper then deals with the proj~x-~tiveinterpretation of the Golay codes; these codes are shown to describe Coxeter's configuration in PG(5, 3) and Todd's configuration in PG(11, 2) when interpreted projectively. We obtain two twelve-dimensional representations of M24. One is obtained as the coUineation group that permutes the twelve special points in PC,(11, 2); the other arises by interpreting geometrically the automorphism group of the binary Golay code. Both representations are reducible to eleven-dimensional representations of M24. Keywords. Geometry; Mathieu groups; Golay codes; Coxeter's configuration; hemi- icosahedron; octastigms; dodecastigms. 1. Introduction We shall first review some of the known properties of the Mathieu groups and their linear fractional subgroups. This is mainly a brief resum6 of the first two of Conway's 'Three Lectures on Exceptional Groups' [3] and will serve to establish the notation and underlying concepts of the subsequent sections. The six points of the projective line PL(5) can be labelled by the marks of GF(5) together with a sixth symbol oo defined to be the inverse of 0. The symbol set is f~ = {0 1 2 3 4 oo}. -
Spanning Sets and Scattering Sets in Steiner Triple Systems
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF COMBINATORIAL THEORY, %XieS A 57, 4659 (1991) Spanning Sets and Scattering Sets in Steiner Triple Systems CHARLESJ. COLBOURN Department of Combinatorics and Optimization, University of Waterloo, Waterloo. Ontario, N2L 3G1, Canada JEFFREYH. DMTZ Department of Mathematics, University of Vermont, Burlington, Vermont 05405 AND DOUGLASR. STINSON Department of Computer Science, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada Communicated by the Managing Editors Received August 15, 1988 A spanning set in a Steiner triple system is a set of elements for which each element not in the spanning set appears in at least one triple with a pair of elements from the spanning set. A scattering set is a set of elements that is independent, and for which each element not in the scattering set is in at most one triple with a pair of elements from the scattering set. For each v = 1,3 (mod 6), we exhibit a Steiner triple system with a spanning set of minimum cardinality, and a Steiner triple system with a scattering set of maximum cardinality. In the process, we establish the existence of Steiner triple systems with complete arcs of the minimum possible cardinality. 0 1991 Academic Press, Inc. 1. BACKGROUND A Steiner triple system of order u, or STS(u), is a v-element set V, and a set g of 3-element subsets of V called triples, for which each pair of elements from V appears in precisely one triple of 9% For XG V, the spannedsetV(X)istheset {~EV\X:{~,~,~)E~,~,~EX}.A~~~XEV 46 0097-3165/91 $3.00 Copyright 0 1991 by Academic Press, Inc. -
Weak Orientability of Matroids and Polynomial Equations
WEAK ORIENTABILITY OF MATROIDS AND POLYNOMIAL EQUATIONS J.A. DE LOERA1, J. LEE2, S. MARGULIES3, AND J. MILLER4 Abstract. This paper studies systems of polynomial equations that provide information about orientability of matroids. First, we study systems of linear equations over F2, originally alluded to by Bland and Jensen in their seminal paper on weak orientability. The Bland-Jensen linear equations for a matroid M have a solution if and only if M is weakly orientable. We use the Bland-Jensen system to determine weak orientability for all matroids on at most nine elements and all matroids between ten and twelve elements having rank three. Our experiments indicate that for small rank, about half the time, when a simple matroid is not orientable, it is already non-weakly orientable, and further this may happen more often as the rank increases. Thus, about half of the small simple non-orientable matroids of rank three are not representable over fields having order congruent to three modulo four. For binary matroids, the Bland-Jensen linear systems provide a practical way to check orientability. Second, we present two extensions of the Bland-Jensen equations to slightly larger systems of non-linear polynomial equations. Our systems of polynomial equations have a solution if and only if the associated matroid M is orientable. The systems come in two versions, one directly extending the Bland-Jensen system for F2, and a different system working over other fields. We study some basic algebraic properties of these systems. Finally, we present an infinite family of non-weakly-orientable matroids, with growing rank and co-rank.