DOCSLIB.ORG

Finite Incidence Geometry in GAP

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Finite Incidence Geometry in GAP Finite Incidence Geometry in GAP GAP in Algebraic Research Jan De Beule 21 November 2018 Contents Introduction Projective spaces Incidence Geometry Projective geometry over (skew)fields Groups of collineations of a projective space Polar spaces Geometry morphisms Generalized polygons Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 1/55 I Jacques Tits, geometry of finite simple groups. I many others .... Some history on (finite) incidence geometry I Beniamino Segre, classical algebraic geometry. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 2/55 I many others .... Some history on (finite) incidence geometry I Beniamino Segre, classical algebraic geometry. I Jacques Tits, geometry of finite simple groups. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 2/55 Some history on (finite) incidence geometry I Beniamino Segre, classical algebraic geometry. I Jacques Tits, geometry of finite simple groups. I many others .... Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 2/55 FinInG I Finite Incidence Geometry in GAP. I Development started in 2005. I Authors: John Bamberg, Anton Betten, Philippe Cara, Jan De Beule, Michel Lavrauw, and Max Neunhöffer I http://www.fining.org I https://bitbucket.org/jdebeule/fining I https://www.gap-system.org/Packages/ fining.html I support ’at’ fining.org Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 3/55 Projective spaces I Let K be a (skew)field. I Let V(n + 1; K) be a (left) vector space over K of dimension n + 1. Definition A projective space of dimension n over the skewfield K is the geometry consisting of the sub vector spaces of V(n + 1; K), with incidence being symmetrized set theoretic containment. We denote this projective space as PG(n; K). The points, lines, planes, ..., hyperplanes, respectively, are the 1-, 2-, 3-, ..., n-dimensional subspaces of V(n + 1; K). Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 4/55 My first projective space in GAP Figure: Load FinFinG Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 5/55 I Points I Lines I List I in My first projective space in GAP I GF I ProjectiveSpace, alternatively PG Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 6/55 I in My first projective space in GAP I GF I ProjectiveSpace, alternatively PG I Points I Lines I List Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 6/55 My first projective space in GAP I GF I ProjectiveSpace, alternatively PG I Points I Lines I List I in Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 6/55 My first projective space in GAP Figure: Fano plane in FinInG, file: myfirstpg.g Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 7/55 Incidence geometry Definition (based on e.g. [2]) An incidence structure is a tuple (S; T; t; I), such that I S a non empty set called the set of elements, I the set T is a finite, non empty set, called the set of types, I t : S! T, is a function and for x 2 S, t(x) is called its type, I the relation I is a symmetric relation on S, called the incidence relation, I two elements of the same type are never incident. The rank of an incidence structure (S; T; t; I) is jTj, the cardinality of T. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 8/55 Incidence geometry Definition A flag of an incidence structure (S; T; I) is a set F ⊂ S such that every two elements of F are incident. From the definition it follows immediately that a flag contains at most one element of a given type. Definition A flag F is maximal if it is not contained in a larger flag. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 9/55 Incidence geometry Definition An incidence structure is called an incidence geometry if every maximal flag contains an element of each type. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 10/55 A projective space as incidence geometry I Rank I TypesOfElementsOfIncidenceStructure I ElementsOfIncidenceStructure I Points, Lines, Planes, Solids, Hyperplanes I IsIncident Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 11/55 A projective space as incidence geometry file: pgincidence1.g Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 12/55 A projective space as incidence geometry file: pgincidence2.g Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 13/55 Some examples I The projective space PG(n; K) is an incidence geometry of rank n. I Define I S = f1; 2; 3; 4; 5; 6; 7g [ ff1; 2; 3g; f1; 4; 5g; f1; 6; 7g; f2; 4; 6g; f3; 4; 7g; f2; 5; 7g; f3; 5; 6gg, I T = f“integer”,“set of integers”g, I t is the obvious function from S to T, I the incidence relation I is symmetrized containment. Then (S; T; t; I) is an incidence geometry of rank 2. We could have called elements of type “integer” points as well, and elements of type “set of integers” lines as well, and vice versa. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 14/55 Graph theoretical point of view I An finite incidence structure of rank r is a multipartite graph (r partitions), together with a type function on the set of vertices and where the incidence relation is simply the adjacency relation of the graph. I Given a finite incidence structure, it is easy to create this incidence graph. I The flags are the cliques of the graph. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 15/55 An incidence structure in FinInG file: incstructure.g Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 16/55 Elements and objects I An element of an incidence geometry is represented by an object. I VectorSpaceToElement and UnderlyingObject. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 17/55 Elements and objects file: vectorspacetoelement.g Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 18/55 Elements and shadows Definition The shadow of type i of an element, respectively flag, is the set of elements of type i incident with the element, respectively with all elements of the flag. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 19/55 Elements and shadows file: shadow.g Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 20/55 More examples of incidence geometries . ...available in FinInG. I Coset geometries I Affine spaces. I Finite classical polar spaces. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 21/55 Desarguesian projective spaces Theorem Let K be a skewfield and n ≥ 2. The projective geometry PG(n; K) is Desarguesian. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 22/55 Desarguesian projective spaces Figure: The theorem of Desargues, figure taken from [1] . Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 23/55 Pappian projective spaces Theorem Let K be a field and n ≥ 2. The projective geometry PG(n; K) is Pappian. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 24/55 Pappian projective spaces Figure: The theorem of Pappus, figure taken from [1] . Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 25/55 Axiomatic projective planes Definition An axiomatic projective plane P is a point line geometry satisfying the following axioms I Any two points determine a unique line. I Any two lines meet in exactly one point. I There are four points of which no three are collinear. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 26/55 Axiomatic projective spaces Definition An axiomatic projective space P is a point line geometry satisfying the following axioms I Any two points determine a unique line. I (Veblen-Younga) Let A, B, C, and D be four points such that the line AB intersects the line CD. Then the line AC also intersects the line BD. I Any line is incident with at least three points. I There are at least two lines. asometimes also called the axiom of Pasch Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 27/55 Axiomatic projective spaces Figure: The axiom of Veblen-Young, figure taken from [1] . Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 28/55 Classical theorems Theorem A projective space P is Desarguesian if and only if P = PG(n; K) for some n ≥ 2 and K a skewfield. A projective space P of dimension at least 3 is Desarguesian. Theorem A projective space P is Pappian if and only if P = PG(n; K) for some n ≥ 2 and K a field. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 29/55 One more classical theorem Theorem (Hessenberg) If the theorem of Pappus hold in the projective space P, then also the theorem of Desargues. Jan De Beule Finite Incidence Geometry in GAP 21 November 2018 30/55 Classical theorems Corollary (by Wedderburn) A finite projective space P is Pappian if and only if it is Desarguesian. Theorem (Fundamental theorem of projective geom- etry) All the collineations of PG(n; K) are induced by the group of semilinear bijective maps of the underlying vector space. A finite projective space PG(n; K), K = Fq, will be denoted PG(n; q).
Load more

Recommended publications
  • Study Guide for the Midterm. Topics: • Euclid • Incidence Geometry • Perspective Geometry • Neutral Geometry – Through Oct.18Th
    Study guide for the midterm. Topics: • Euclid • Incidence geometry • Perspective geometry • Neutral geometry { through Oct.18th. You need to { Know definitions and examples. { Know theorems by names. { Be able to give a simple proof and justify every step. { Be able to explain those of Euclid's statements and proofs that appeared in the book, the problem set, the lecture, or the tutorial. Sources: • Class Notes • Problem Sets • Assigned reading • Notes on course website. The test. • Excerpts from Euclid will be provided if needed. • The incidence axioms will be provided if needed. • Postulates of neutral geometry will be provided if needed. • There will be at least one question that is similar or identical to a homework question. Tentative large list of terms. For each of the following terms, you should be able to write two sentences, in which you define it, state it, explain it, or discuss it. Euclid's geometry: - Euclid's definition of \right angle". - Euclid's definition of \parallel lines". - Euclid's \postulates" versus \common notions". - Euclid's working with geometric magnitudes rather than real numbers. - Congruence of segments and angles. - \All right angles are equal". - Euclid's fifth postulate. - \The whole is bigger than the part". - \Collapsing compass". - Euclid's reliance on diagrams. - SAS; SSS; ASA; AAS. - Base angles of an isosceles triangle. - Angles below the base of an isosceles triangle. - Exterior angle inequality. - Alternate interior angle theorem and its converse (we use the textbook's convention). - Vertical angle theorem. - The triangle inequality. - Pythagorean theorem. Incidence geometry: - collinear points. - concurrent lines. - Simple theorems of incidence geometry. - Interpretation/model for a theory.
    [Show full text]
  • 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.
    [Show full text]
  • Arxiv:1609.06355V1 [Cs.CC] 20 Sep 2016 Outlaw Distributions And
    Outlaw distributions and locally decodable codes Jop Bri¨et∗ Zeev Dvir† Sivakanth Gopi‡ September 22, 2016 Abstract Locally decodable codes (LDCs) are error correcting codes that allow for decoding of a single message bit using a small number of queries to a corrupted encoding. De- spite decades of study, the optimal trade-off between query complexity and codeword length is far from understood. In this work, we give a new characterization of LDCs using distributions over Boolean functions whose expectation is hard to approximate (in L∞ norm) with a small number of samples. We coin the term ‘outlaw distribu- tions’ for such distributions since they ‘defy’ the Law of Large Numbers. We show that the existence of outlaw distributions over sufficiently ‘smooth’ functions implies the existence of constant query LDCs and vice versa. We also give several candidates for outlaw distributions over smooth functions coming from finite field incidence geometry and from hypergraph (non)expanders. We also prove a useful lemma showing that (smooth) LDCs which are only required to work on average over a random message and a random message index can be turned into true LDCs at the cost of only constant factors in the parameters. 1 Introduction Error correcting codes (ECCs) solve the basic problem of communication over noisy channels. They encode a message into a codeword from which, even if the channel partially corrupts it, arXiv:1609.06355v1 [cs.CC] 20 Sep 2016 the message can later be retrieved. With one of the earliest applications of the probabilistic method, formally introduced by Erd˝os in 1947, pioneering work of Shannon [Sha48] showed the existence of optimal (capacity-achieving) ECCs.
    [Show full text]
  • Connections Between Graph Theory, Additive Combinatorics, and Finite
    UNIVERSITY OF CALIFORNIA, SAN DIEGO Connections between graph theory, additive combinatorics, and finite incidence geometry A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Michael Tait Committee in charge: Professor Jacques Verstra¨ete,Chair Professor Fan Chung Graham Professor Ronald Graham Professor Shachar Lovett Professor Brendon Rhoades 2016 Copyright Michael Tait, 2016 All rights reserved. The dissertation of Michael Tait is approved, and it is acceptable in quality and form for publication on microfilm and electronically: Chair University of California, San Diego 2016 iii DEDICATION To Lexi. iv TABLE OF CONTENTS Signature Page . iii Dedication . iv Table of Contents . .v List of Figures . vii Acknowledgements . viii Vita........................................x Abstract of the Dissertation . xi 1 Introduction . .1 1.1 Polarity graphs and the Tur´annumber for C4 ......2 1.2 Sidon sets and sum-product estimates . .3 1.3 Subplanes of projective planes . .4 1.4 Frequently used notation . .5 2 Quadrilateral-free graphs . .7 2.1 Introduction . .7 2.2 Preliminaries . .9 2.3 Proof of Theorem 2.1.1 and Corollary 2.1.2 . 11 2.4 Concluding remarks . 14 3 Coloring ERq ........................... 16 3.1 Introduction . 16 3.2 Proof of Theorem 3.1.7 . 21 3.3 Proof of Theorems 3.1.2 and 3.1.3 . 23 3.3.1 q a square . 24 3.3.2 q not a square . 26 3.4 Proof of Theorem 3.1.8 . 34 3.5 Concluding remarks on coloring ERq ........... 36 4 Chromatic and Independence Numbers of General Polarity Graphs .
    [Show full text]
  • 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].
    [Show full text]
  • 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.
    [Show full text]
  • On the Theory of Point Weight Designs Royal Holloway
    On the theory of Point Weight Designs Alexander W. Dent Technical Report RHUL–MA–2003–2 26 March 2001 Royal Holloway University of London Department of Mathematics Royal Holloway, University of London Egham, Surrey TW20 0EX, England http://www.rhul.ac.uk/mathematics/techreports Abstract A point-weight incidence structure is a structure of blocks and points where each point is associated with a positive integer weight. A point-weight design is a point-weight incidence structure where the sum of the weights of the points on a block is constant and there exist some condition that specifies the number of blocks that certain sets of points lie on. These structures share many similarities to classical designs. Chapter one provides an introduction to design theory and to some of the existing theory of point-weight designs. Chapter two develops a new type of point-weight design, termed a row-sum point-weight design, that has some of the matrix properties of classical designs. We examine the combinatorial aspects of these designs and show that a Fisher inequality holds and that this is dependent on certain combinatorial properties of the points of minimal weight. We define these points, and the designs containing them, to be either ‘awkward’ or ‘difficult’ depending on these properties. Chapter three extends the combinatorial analysis of row-sum point-weight designs. We examine structures that are simultaneously row-sum and point-sum point-weight designs, paying particular attention to the question of regularity. We also present several general construction techniques and specific examples of row-sum point-weight designs that are generated using these techniques.
    [Show full text]
  • Keith E. Mellinger
    Keith E. Mellinger Department of Mathematics home University of Mary Washington 1412 Kenmore Avenue 1301 College Avenue, Trinkle Hall Fredericksburg, VA 22401 Fredericksburg, VA 22401-5300 (540) 368-3128 ph: (540) 654-1333, fax: (540) 654-2445 [email protected] http://www.keithmellinger.com/ Professional Experience • Director of the Quality Enhancement Plan, 2013 { present. UMW. • Professor of Mathematics, 2014 { present. Department of Mathematics, UMW. • Interim Director, Office of Academic and Career Services, Fall 2013 { Spring 2014. UMW. • Associate Professor, 2008 { 2014. Department of Mathematics, UMW. • Department Chair, 2008 { 2013. Department of Mathematics, UMW. • Assistant Professor, 2003 { 2008. Department of Mathematics, UMW. • Research Assistant Professor (Vigre post-doc), 2001 - 2003. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL, 60607-7045. Position partially funded by a VIGRE grant from the National Science Foundation. Education • Ph.D. in Mathematics: August 2001, University of Delaware, Newark, DE. { Thesis: Mixed Partitions and Spreads of Projective Spaces • M.S. in Mathematics: May 1997, University of Delaware. • B.S. in Mathematics: May 1995, Millersville University, Millersville, PA. { minor in computer science, option in statistics Grants and Awards • Millersville University Outstanding Young Alumni Achievement Award, presented by the alumni association at MU, May 2013. • Mathematical Association of America's Carl B. Allendoerfer Award for the article,
    [Show full text]
  • 1 Definition and Models of Incidence Geometry
    1 Definition and Models of Incidence Geometry (1.1) Definition (geometry) A geometry is a set S of points and a set L of lines together with relationships between the points and lines. p 2 1. Find four points which belong to set M = f(x;y) 2 R jx = 5g: 2. Let M = f(x;y)jx;y 2 R;x < 0g: p (i) Find three points which belong to set N = f(x;y) 2 Mjx = 2g. − 4 L f 2 Mj 1 g (ii) Are points P (3;3), Q(6;4) and R( 2; 3 ) belong to set = (x;y) y = 3 x + 2 ? (1.2) Definition (Cartesian Plane) L Let E be the set of all vertical and non-vertical lines, La and Lk;n where vertical lines: f 2 2 j g La = (x;y) R x = a; a is fixed real number ; non-vertical lines: f 2 2 j g Lk;n = (x;y) R y = kx + n; k and n are fixed real numbers : C f 2 L g The model = R ; E is called the Cartesian Plane. C f 2 L g 3. Let = R ; E denote Cartesian Plane. (i) Find three different points which belong to Cartesian vertical line L7. (ii) Find three different points which belong to Cartesian non-vertical line L15;p2. C f 2 L g 4. Let P be some point in Cartesian Plane = R ; E . Show that point P cannot lie simultaneously on both L and L (where a a ). a a0 , 0 (1.3) Definition (Poincar´ePlane) L Let H be the set of all type I and type II lines, aL and pLr where type I lines: f 2 j g aL = (x;y) H x = a; a is a fixed real number ; type II lines: f 2 j − 2 2 2 2 g pLr = (x;y) H (x p) + y = r ; p and r are fixed R, r > 0 ; 2 H = f(x;y) 2 R jy > 0g: H f L g The model = H; H will be called the Poincar´ePlane.
    [Show full text]
  • Second Edition Volume I
    Second Edition Thomas Beth Universitat¨ Karlsruhe Dieter Jungnickel Universitat¨ Augsburg Hanfried Lenz Freie Universitat¨ Berlin Volume I PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK www.cup.cam.ac.uk 40 West 20th Street, New York, NY 10011-4211, USA www.cup.org 10 Stamford Road, Oakleigh, Melbourne 3166, Australia Ruiz de Alarc´on 13, 28014 Madrid, Spain First edition c Bibliographisches Institut, Zurich, 1985 c Cambridge University Press, 1993 Second edition c Cambridge University Press, 1999 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1999 Printed in the United Kingdom at the University Press, Cambridge Typeset in Times Roman 10/13pt. in LATEX2ε[TB] A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data Beth, Thomas, 1949– Design theory / Thomas Beth, Dieter Jungnickel, Hanfried Lenz. – 2nd ed. p. cm. Includes bibliographical references and index. ISBN 0 521 44432 2 (hardbound) 1. Combinatorial designs and configurations. I. Jungnickel, D. (Dieter), 1952– . II. Lenz, Hanfried. III. Title. QA166.25.B47 1999 5110.6 – dc21 98-29508 CIP ISBN 0 521 44432 2 hardback Contents I. Examples and basic definitions .................... 1 §1. Incidence structures and incidence matrices ............ 1 §2. Block designs and examples from affine and projective geometry ........................6 §3. t-designs, Steiner systems and configurations .........
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]