Table Of ContentSpringer 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
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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
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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 combina-
torics 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 com-
binatorial problems.
Geometry has often formed the basis for the development of the rest of math-
ematics.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 math-
ematics. 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 recre-
ational 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 recre-
ational 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
