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View Is That the Family of Subsets Is in Fact a Multiset of the Set of All K-Subsets of X and One May Well Have a Multiple Occurence (Or Repetition) of a Block THE 3−DESIGN PROBLEM DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Niranjan Balachandran, M. Stat(India) ***** The Ohio State University 2008 Dissertation Committee: Approved by Professor Neil Robertson, Advisor Professor Akos´ Seress Advisor Professor Stephen Milne Graduate Program in Mathematics ABSTRACT The theory of Combinatorial designs is one of the oldest and richest disciplines in Combinatorics and has wide ranging applications in as diverse fields as Cryptography, Optics, Discrete Tomography, data structures and computer algorithms, hardware design, Interconnection networks, VLSI testing, Astronomical Imaging, and Neutron Spectroscopy and also contributes to other disciplines of Mathematics such as The theory of Unimodular lattices, Coding Theory, Computational Group theory, and Discrete and Combinatorial Geometry. A t−(v, k, λ) design is a pair (X, B), where X is a set of size v and B is a collection of subsets of X of size k each such that every t-subset of X is contained in precisely λ members of B.A t − (v, k, λ) design is also denoted Sλ(t, k, v). If λ = 1 then it is called a t-Steiner design and is denoted by S(t, k, v). The problem of characterizing all triples (v, k, λ) for which a t − (v, k, λ) design exists is one of the fundamental problems in the theory of Combinatorial designs. Clearly, these parameters cannot be chosen independent of one another since there are certain necessary conditions that are to be met in order that a design exists. These are known as the arithmetic conditions or the ‘admissibility conditions’. While the admissibility conditions are necessary, they are also not sufficient; there exist several cases of parameters that satisfy the admissibility conditions and yet no ii design with these parameters exists. However, if the point set is large, then it is conjectured that the admissibility conditions would be sufficient as well. This is known as the ‘v-large existence conjecture’ or the ‘asymptotic existence’ conjecture. The ‘asymptotic existence’ conjecture has been proved for t = 2 by Wilson, following the work of several including R.C. Bose, Marshall Hall, Jr., Haim Hanani, and Dijen Ray-Chaudhuri. This dissertation studies the ‘asymptotic existence’ conjecture in the specific case t = 3 with the primary goal of constructing new families of 3-designs . More specifi- cally, this dissertation includes the following: 1. Firstly, by considering the action of the group PSL(2, q) on the finite projective line and the orbits of the action of this group to construct simple 3-designs. While the case q ≡ 3 (mod 4) is 3-homogeneous (so that orbits of any ‘base’ block’ would yield designs), the case q ≡ 1 (mod 4) does not work the same way. We however overcome some of these issues by considering appropriate unions of orbits to produce new infinite families of 3-designs with PSL(2, q) acting as a group of automorphisms. We also prove that our constructions actually produce an abundance of simple 3-designs for any block size if q is sufficiently large. We also construct a large set of Divisible designs as an application of our constructions. 2. We generalize the notion of a Candelabra system to more general structures, called Rooted Forest Set systems and prove a few general results on combina- torial constructions for these general set structures. Then, we specialize to the iii case of k = 6 and extend a theorem of Hanani to produce several new infinite families of Steiner 3-designs with block size 6. 3. Finally, we consider Candelabra systems and prove that a related incidence ma- trix has full row rank over Q. This leads to interesting possibilities for λ large theorems for Candelabra systems. While a λ-large theorem for Candelabra sys- tems do not directly yield any Steiner 3-design (in fact, even simple 3 designs), it allows for constructions of new Steiner 3-designs on large sets following the methods of Block spreading. iv To U. Koteswara Rao(1948-2008) - the first real mathematician I knew personally. v ACKNOWLEDGMENTS No one who achieves success does so without acknowledging the help of others. The wise and confident acknowledge this help with gratitude. Alfred North Whitehead. It is very difficult to make a complete list of all the people I am indebted to, since that would fill more pages than I can imagine. First and foremost goes my gratitude towards Professor Dijen Ray-Chaudhuri. He has been more than a mentor to me. Apart from imparting to me a great deal of knowledge on a subject in which he is indeed a renowned expert, he taught me to think like a design theorist which is more important in the long run. He was kind even towards some of my silliest ideas and always had an encouraging word, especially when I felt I was getting nowhere. My thanks to the members of my thesis committee cannot be underemphasized. I am indeed grateful to Dr Neil Robertson, Dr. Akos´ Seress, and Dr. Stephen Milne for all their help. I am also extremely thankful to Dr. Arasu for several useful mathematical dis- cussions, his general advice, and great food at his place! vi My interest in math came about would take me back to the time I was in ele- mentary school. However, some of the teachers I have had, were instrumental in my desire to make a career out of being a mathematician. Indeed, one of my earliest math teachers Mr U. Koteswara Rao, made a mathematician look cool! I also owe a lot of my mathematical training to my alma mater, The Indian Statistical Institute, India. My 5 year stint at ISI, along with the summer camps at TIFR, Mumbai, led me to believe that a career in mathematics was what I had to pursue. In such a situation there are always several others who deserve more than a mere thank-you. My parents have always been extremely supportive of anything I did; my mom has spent several an occasion listening to me describing the ‘work I do everyday’, in an appreciative tone when, I am sure, she had no idea what I was talking about! My sisters Neeraja and Sindu have always been among my best friends, always encouraging, especially when I found myself in a slump. Among my close friends I particularly thank Rajeshpavan, Harish, Prasenjit, and Gopal for being a great help any time and every time! Lastly, and by no means least, my gratitude to my wife Anupama. Anu has been a great support, ever helpful and cheerful, and has shown a lot of patience towards me and my irregular life style. I am inclined to think my fortunes have changed for the better ever since she became a part of my life. vii VITA December 24, 1977 . Born - Vellore, Tamil Nadu, India 1996-1999 . B. Stat(Hons), Indian Statistical Institute, Kolkata, India 1999-2001 . M. Stat., Indian Statistical Institute, Bangalore, India 2001-present . Graduate Teaching Associate, The Ohio State University PUBLICATIONS 1. “Simple 3-designs and PSL(2, q) with q ≡ 1 (mod 4) ”, with Dijen Ray-Chaudhuri, Designs, Codes and Cryptography, 44(2007), 263-274. 2. “Graphs with restricted valency and matching number”, with Niraj Khare, submitted to Discrete Math (preprint available in arXiv.org :math/0611842v1), 2006. 3. “New infinite families of Candelabra Systems with block size 6 and stem size 2 ”, submitted, 2007. 4. “ A λ-large theorem for Candelabra Systems”, in preparation, 2007. viii FIELDS OF STUDY Major Field: Mathematics Specialization: Combinatorial Design theory, Finite Geometry ix TABLE OF CONTENTS Abstract . ii Dedication . v Acknowledgments . vi Vita . viii CHAPTER PAGE 1 The 3-design problem: General theory and results . 1 1.1 What is a Design? . 1 1.2 Steiner 2-designs and Finite Geometry . 5 1.3 Problems in Design Theory . 11 1.4 The problem of existence . 12 1.5 3-designs . 15 1.6 Algebraic constructions for 3-designs . 17 1.7 Combinatorial constructions for 3-designs . 19 1.8 λ-large theorems . 23 2 Algebraic Constructions . 27 2.1 Classical 3-transitive groups . 28 2.2 The Groups PSL(2, q), q odd................ 38 2.3 The case q ≡ 3 (mod 4) . 43 2.4 The case q ≡ 1 (mod 4) . 46 2.5 Large sets of 3-DDs . 62 2.6 Concluding remarks . 65 3 Combinatorial Constructions . 67 3.1 Introduction . 67 3.2 Candelabra systems and Rooted Forest Set Systems(RFSS) 68 x 3.3 General constructions . 74 3.4 New constructions for k = 6 from Candelabra systems . 82 3.5 New constructions for k = 6: The Proofs . 85 3.6 Concrete instances of the theorems . 92 3.7 Steiner designs with block size 5 . 94 3.8 Concluding Remarks . 95 4 ‘λ-large’ theorems . 97 4.1 Preliminaries . 98 4.2 Proof of the Theorem . 101 4.3 Concluding remarks . 109 Bibliography . 111 xi CHAPTER 1 THE 3-DESIGN PROBLEM: GENERAL THEORY AND RESULTS If you would hit the mark, you must aim a little above it. Henry Wadsworth Longfellow. 1.1 What is a Design? The notion of a Combinatorial design can be traced back to the Rev. T. P. Kirkman (1806-1895) who first posed a puzzle in Lady’s and Gentleman’s Diary in 1845 which has since then come to be known as the ‘15 Schoolgirl Problem’: “Fifteen young ladies of a school walk out three abreast for seven days in succession: it is required to arrange them daily so that no two shall walk abreast more than once..” Trial-and-error methods would quickly convince anyone that a solution to this prob- lem is by no means trivial.
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