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Springer Undergraduate Mathematics Series Steven T. Dougherty Combinatorics and Finite Geometry Springer Undergraduate Mathematics Series Advisory Editors Mark A. J. Chaplain, St. Andrews, UK Angus Macintyre, Edinburgh, UK Simon Scott, London, UK Nicole Snashall, Leicester, UK Endre Süli, Oxford, UK Michael R. Tehranchi, Cambridge, UK John F. Toland, Bath, UK The Springer Undergraduate Mathematics Series (SUMS) is a series designed for undergraduatesinmathematicsandthesciencesworldwide.Fromcorefoundational material to final year topics, SUMS books take a fresh and modern approach. Textual explanations are supported by a wealth of examples, problems and fully-workedsolutions,withparticularattentionpaidtouniversalareasofdifficulty. Thesepracticalandconcisetextsaredesignedforaone-ortwo-semestercoursebut the self-study approach makes them ideal for independent use. More information about this series at http://www.springer.com/series/3423 Steven T. Dougherty Combinatorics and Finite Geometry 123 StevenT. Dougherty Department ofMathematics University of Scranton Scranton, PA,USA ISSN 1615-2085 ISSN 2197-4144 (electronic) SpringerUndergraduate MathematicsSeries ISBN978-3-030-56394-3 ISBN978-3-030-56395-0 (eBook) https://doi.org/10.1007/978-3-030-56395-0 MathematicsSubjectClassification: 05,06,51,52,94 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature SwitzerlandAG2020 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseof illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. 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Preface The mathematics that will be discussed in this text falls into the branch of math- ematicsknownascombinatorics.Combinatoricsisaveryoldandverylargebranch of mathematics encompassing numerous topics. One nice feature about combinatorics is that you need to know very little mathematics to understand the open questions. Unlike many branches of mathematics, there is very little technical languageandveryfewdefinitionsstandinginthewayofunderstandinginteresting unsolved problems in the area. At present, there are numerous problems which no onehasansweredyet,butthatcanbeexplainedtoalmostanyone.Infact,amateurs have, on occasion, made interesting discoveries in combinatorics. We shall talk about one such example in the third chapter. Ontheotherhand,theseopenproblemsareoftenvery,very,difficulttosolve.In astrangetwist,itisoftentruethattheeasieritistostateaproblem,theharderitis to solve it. Many problems in mathematics or science seem hopelessly difficult because it requires a good deal of study just to understand the language of the problem. However, it is often true that the standard techniques of the field can be appliedtosolvetheproblem.Soeventhoughtheproblemsmayseemverydifficult, they are not. When problems are easy to state and have been around for a while, then you can be fairly certain that the standard techniques do not apply. These problems are like infections for which known antibiotics simply do not work. To solvethemyouneedsomethingnewandoriginal.Inthissamevein,theyareoften very infectious, once you begin to think about them you crave to know their solution. In a very real sense, the best in mathematics is like unrequited love, the more the discipline refuses to reveal its secrets the more you desire them. Combinatorics is also interesting because it has a wide overlap with other branchesofmathematicsincludingabstractalgebra,numbertheory,codingtheory, andtopology.Theseandotherbranchesofmathematicsoftenhavetechniques that areusefulinsolvingtheproblemsthatariseincombinatoricsandoftenproblemsin theseotherareashavecombinatorialsolutions.Combinatoricsalsohasaverywide varietyofapplicationsinscience.Infact,manypartsofcombinatorics,whichwere purely abstract at their birth, were later vital to scientific applications. In fact, the dawn of the electronic age has brought forth a great interest in combinatorial matters because of their connection to computers and the communication of information. vii viii Preface While combinatorics is a very rich and diverse subject, we shall focus our attention to some specific areas, for example, finite geometry and design theory. Finitegeometry,asthenamesuggests,iscloselyrelatedtoinfinitegeometry.They sharemanyofthesameideasanddefinitions.Geometry,alongwithnumbertheory, is one of the oldest branches of mathematics. In every culture that developed mathematics, geometry developed first. The reason for this is clear. Namely, geometry is probably the most intuitive branch of mathematics. The ideas behind geometry are very natural to consider and have applications from the most ele- mentary aspects of life to the most advanced science. In finite geometry, we study these geometric ideas in a finite setting and relate these intuitive notions to combinatorial problems. Geometry has often formed the basis for the development of the rest of mathematics.Modernmathematics wasbuiltonthebaseofancientGreekmathematics, which phrased every idea (even algebraic ones) in terms of geometry. Greek mathematics greatly influenced the focus, notation, and tone of all modern mathematics. It was on this intuitive, geometric foundation that mathematics was built. In the nineteenth century, mathematicians began to look at all mathematics including geometry in a very abstract way and created models for different geometries using algebraic methods. It was noticed that these same techniques could be used to create geometries, which were finite in that they contained only finitely many points and lines. These are geometries where many of the funda- mental properties of standard geometry were true, but there were only a finite numberofpointsandlines.Itwassoonrealizedthatfinitegeometryhadnumerous connections with interesting questions from combinatorics. Design theory arose fromthisconnection.Adesigncanbethoughtofasageometricunderstandingofa combinatorial problem. Later,inthesecondhalfofthetwentiethcentury,itwasrealizedthatthesevery abstract ideas in finite geometry could be used in the very concrete problem of electronic communication. Many of the ideas in this text have applications both in themathematicsofcodingtheory(makingsurethatamessageisreceivedcorrectly) and in cryptography (making sure that secret messages are not read by undesired recipients).Onesmallexampleofhowfinitegeometrycanbeusedincodingtheory will be shown in the text. Numerous other applications of combinatorics exist in mathematics, statistics, and science. There has always been a healthy exchange of ideas from those who study combinatorics as pure mathematics and others who apply it. Namely, those who apply it give the researcher interesting problems to thinkabout,andtheresearchergivessolutionstothecombinatorialquestionsraised by others. Many combinatorial problems, on the other hand, have their origins in recreational mathematics and from questions arising in other branches of mathematics. Often, some interesting questions in mathematics have arisen in very odd cir- cumstances. For example, the Kirkman schoolgirl problem was raised as a recreational problem as was the question of the existence of the mutually orthogonal Latin squares. We shall give some examples, including these, in the text of how Preface ix some interesting recreational mathematics gave rise to some of the most well-known and important questions in combinatorics. Notations We shall establish some notations that will be used throughout the text. We begin withthenaturalnumbers.WeshalldenotethenaturalnumbersbyNandweassume that 0 is a natural number. Therefore, N¼f0;1;2;3;...g: We denote the integers byZandZ¼f...;(cid:2)3;(cid:2)2;(cid:2)1;0;1;2;3;...g:WedenotetherationalsbyQwhere Q¼faja;b2Z;b6¼0g: We denote the greatest integer function of x by bxc, b namely, bxc¼maxfnjn(cid:3)x;n2Z}. Guide for Lecturers An effort has been made to make each chapter independent and self-contained to allow it to be read without having to refer back constantly to earlier chapters. The topics in Sects. 1.1–1.4, 2.1, and 2.2 are really the only ones that must be under- stoodtoreadtheremainingchapters.(Itshouldbepointedoutthatwhilethemajor results of Sect. 2.2, for example, the existence of finite fields of all prime power orders, must be understood, it is not necessary to have a complete understanding of the algebraic techniques used to obtain these results.) Given a knowledge of these basic chapters, it should be possible for a student to learn the material independently or to make an interesting course by choosing topics from the remaining sections. It is possible to make several different courses from this text. For students who have not had a course in discrete mathematics, nor a bridge course, the author suggests that Chaps. 1–6 make a nice introductory course on the subject. In this scenario, it is probably best to do the first three sections of Chap. 6. It is also possibletoskipthetopicsfromSect.2.3astheyarenotusedintheremainderofthe text,butitisprobablybesttochooseoneofthesetypesofnumberstoinclude.The topics from Chap. 13 can be introduced throughout to allow students to get some concreteexperiencewiththecombinatorialobjects.Inthiscase,onecanmaketwo courses, where the second course consists of the remaining sections of Chap. 6 along with Chaps. 7–13. This configuration also works well if the first course is a standard course and the second one is an independent study course. Forstudentswhohavehadacourseindiscretemathematicsorabridgecourse,it ispossibletoskipSects.1.1–1.3,and2.1.Giventhatthesestudentsaremorelikely tobemathematicallymature,onecanthenchoosefromtopicsinChaps.7and8or forstudentswhoseinterestisincomputerscienceonemightwishtochoose Chap. 11 or Chap. 12 (or both). x Preface Formoreadvancedstudents,onecanstartwithChap.3andmakeacoursefrom Chaps. 3–10, choosing topics from the remaining 3 chapters based on student interest. Scranton, USA Steven T. Dougherty