Lectures on advances in combinatorics

324 Pages·2010·4.23 MB·English

by  Ahlswede, Rudolf;Blinovskij, Vladimir Markovič

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Rudolf Ahlswede
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Vladimir Blinovsky
Lectures
on Advances
in Combinatorics
1 23

Rudolf Ahlswede · Vladimir Blinovsky
Lectures on Advances
in Combinatorics
RudolfAhlswede VladimirBlinovsky
Universita¨tBielefeld InstituteofInformation
Fakulta¨tfu¨rMathematik TransmissionProblems
Universita¨tsstr.25 RussianAcademyofSciences
33615Bielefeld Bol’shoikaretnyiper.19
Germany 127994Moscow
[email protected] Russia
[email protected]
ISBN978-3-540-78601-6 e-ISBN978-3-540-78602-3
LibraryofCongressControlNumber:2008923540
MathematicsSubjectClassification(2000):05-XX,11-XX,40-XX,52-XX,68-XX,94-XX
(cid:176)c Springer-VerlagBerlinHeidelberg2008
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Preface
The lectures concentrate on highlights in Combinatorial (ChaptersII and III) and
NumberTheoretical(ChapterIV)ExtremalTheory,inparticularonthesolutionof
famousproblemswhichwereopenformanydecades.
However,theorganizationofthelecturesinsixchaptersdoesneitherfollowthe
historicdevelopmentsnortheconnectionsbetweenideasinseveralcases.Withthe
specifiedauxiliaryresultsinChapterIonProbabilityTheory,GraphTheory,etc.,all
chapterscanbereadandtaughtindependentlyofoneanother.
In addition to the 16 lectures organized in 6 chapters of the main part of the
book,thereissupplementarymaterialformostofthemintheAppendix.Inparticu-
lar,thereareapplicationsandfurtherexercises,researchproblems,conjectures,and
evenresearchprograms.
The following books and reports [B97], [ACDKPSWZ00], [A01], and
[ABCABDM06],mostlyoftheauthors,arefrequentlycitedinthisbook,especially
intheAppendix,andwethereforemarkthembyshortlabelsas[B],[N],[E],and
[G].Weemphasizethattherearealso“Exercises”in[B],a“ProblemSection”with
contributions by several authors on pages 1063–1105 of [G], which are often of a
combinatorialnature,and“ProblemsandConjectures”onpages172–173of[E].
The book includes the two well-known results (both in ChapterV),
the Ahlswede/Zhang identity, which improves the LYM-inequality, and the
Ahlswede/Daykin inequality, which is more general and also sharper than known
correlation inequalities in Statistical Physics, Probability Theory, and Combi-
natorics (cf. the survey by Fishburn/Shepp in [N], 501–516). These inequali-
ties were started in Probability Theory (percolation) around 1960 with Harris,
in Combinatorics in 1966 with Kleitman’s Lemma, and in physics 1971 with
Fortuin/Kasteleyn/Ginibre (FKG). In many books the AD-inequality (also called
“4-FunctionTheorem”)isviewedinconnectionwithlattices.Weemphasizethata
muchmoregeneralinequalityof[AD79b]makesnoreferencetolattices.
ItsessenceisaCartesianproductpropertyofsetsandthereforethereisawider
range of possible applications. Also there is nothing holy about the number of
operations and factors on either side of the inequality as long as proper weight-
expansivenessisensured.Inthefollowing,wecometoanothersurpriseconcerning
number-theoreticalinequalities.
v
vi Preface
Aspectacularseriesofresultsstartedwithalecturein1992ofErdo¨s,whoraised
in1962(andrepeatedlyspokeabout)theproblem“Whatisthemaximalcardinality
of a set of numbers smaller than n with no k+1 of its members being pairwise
relativelyprime?”
This stimulated Ahlswede and Khachatrian to make a systematic investiga-
tion of this and related number-theoretical extremal problems. Its immediate suc-
cessesaresolutionsforseveralwell-knownconjecturesofErdo¨sandErdo¨s/Graham
(ChapterVI). More importantly, they gained an understanding for the role of the
primenumberdistributionforsuchproblems,whichdistinguishesthemfromcom-
binatorialextremalproblems.Theseinvestigationshadanothersurprisingfruit.The
AD-inequalityimpliesanumber-theoreticalcorrelationinequalityforDirichletden-
sitiesofsetsofnumbers,whichimpliesandissharperthantheclassicalinequalities
in [H37] and [R37], which settled a conjecture of Hasse concerning an identity
due to Dirichlet and Behrend, the number theoretical form of FKG! Number The-
ory came first and AD is a crossroad between Pure and Applied Mathematics (in
ChapterV). Also another inequality, seemingly without predecessors, was discov-
ered.
Finally, the analysis led to the discovery of a new “pushing” method with a
wide applicability. In particular, it led to the solution of well-known combinator-
ialproblemslikethefamous4m-conjecture(Erdo¨s/Ko/Rado1938,oneoftheoldest
problemsinCombinatorialExtremalTheory)orthediametricprobleminHamming
spaces(optimalanticodes).
Actually,the4m-conjecturejustconcernedthefirstu(cid:1)nso(cid:2)lvedcaseofthefollow-
ingmuchmoregeneralproblem:AsystemofsetsA⊂ [n] iscalledt-intersecting
k
if|A ∩A |≥t forallA ,A ∈A,andI(n,k,t)denotesthesetofallsuchsystems.
1 2 1 2
DeterminethefunctionM(n,k,t)= max |A|andthestructureofmaximalsys-
A∈I(n,k,t)
tems!AhlswedeandKhachatriangavethecompletesolutionforeveryn,k,t.Ithas
averycleargeometricalinterpretation(ChapterII).
Most lectures in ChapterIII are devoted to combinatorics of multiple packings,
which are equivalent to list codes in Information Theory and as such relevant for
estimatingerrorprobabilities.FundamentalworksofBlinovsky[B01a]represented
here deliver the solutions of the problems for list codes, which were stated by the
classicsofinformationtheoryinthemiddleofthelastcentury.Theseproblemsgive
abeautifulexampleofinterplaybetweenExtremalCombinatoricsandInformation
Theory.CoveringandpackingareclassicaltopicsinGeometry.InChapterIIIthey
concernsequencespacesforproblemsprimarilymotivatedbyInformationTheory:
Data Compression and Shannon’s zero error capacity problem of a noisy channel,
whichisapackingproblemforproducthypergraphs.AhighlightwasLova´sz’solu-
tionofthepentagoncase.Ingeneral,theprogressisratherslow.Herewedealwith
arelatedpartitionproblem.
Origins of problems and theories, which developed with them, are only briefly
discussed because of limited space. However, we consider it especially important
forstudentstothinkaboutMathematicsinconnectionwithothersciencesandreal
world phenomena. In fact, a large part of ChaptersIV and V originated that way.
StimulicamemostlyfromquestionsinInformationTheory.Wegiveaquotefrom
[N],pagexvi.
Preface vii
“Thedeepinterplaybetweenseveraldisciplinesandabroadphilosophicalview
isathreadthroughAhlswede’swork.Forhim,InformationTheorydealswithgain-
inginformation(thatis,statistics),transferofinformationwithoutandwithsecrecy
constraints(thatis,cryptology),andstoringinformation(Memories,DataCompres-
sion).Applyingideasfromoneareatoanotheroftenledtounexpectedandbeautiful
resultsandeventonewtheories.
Let us give an example involving storage. Motivated by the practical problem
of storing data using a new laser technique, code models for reusable memories
were introduced in Information Theory. It turned out that the analysis was much
moreefficientwhenstatingthequestionasacombinatorialextremalproblem,which
led immediately to the connections with hypergraph coloring, novel iso-diametric
problemsinsequencespacesandfinallytothenewclassoftheso-called“Higher-
levelextremalproblems”inCombinatorics.
Among them are also Sperner-type questions for “clouds” of antichains. These
problems are by one degree more complex than those usually considered: sets
take the role of elements, families of sets (clouds) take the role of sets, etc. ([N],
P.L.Erdo¨s,L.A.Szekely,117–124).
Inanotherdirection,generalizingmodelsforreusablememories,Ahlswedeand
Zhangintroducedwrite–efficientrewritablememoriesleadingtodiametricalprob-
lemsforsequencespaces.
ImagineatapewithncellsintowhichwecanwritelettersfromanalphabetX.
Awordxn=(x ,...,x )storessomemessages.Whenwewanttoupdatethisrecord
1 n
toamessagerepresentedbyyn=(y ,...,y )theperlettercostsϕ(x,y)addupto
1 n t t
ϕ(xn,yn)=∑n ϕ(x,y).WhenthereisacostconstraintD,thenwerequire
n t=1 t t
ϕ(xn,yn)≤D.
n
Tobeabletoupdatemanymessageswecometothediametricproblemtocharac-
terize
M(ϕ,D)=max{|C|:ϕ(xn,yn)≤D forallxn,yn∈C}
n n
forthe“sum-type”costϕ,whichcanalsobeadistancefunctionliketheHamming,
n
TaxiorLeemetric,etc.TheseproblemsarediscussedalsoinChapterII(andinthe
Appendix).
There, in Lecture 6, also two families of sequences with constant mutual dis-
tances are considered. This falls into the subject of monochromatic rectangles,
whicharoseinYao’sinvestigationofcommunicationcomplexity.
ThetopicofthelastlectureinChapterIIalsohasitsorigininComputerScience
andwascommunicatedtousbyMullinin1990.
Finally, we emphasize that another approach, the study of extremal problems
underdimensionconstraints(inChapterIV)hasfoundanapplicationinComputer
Science.Severalquestionsondatabasesfindanswersasimmediateconsequences.
Anothernewconcept,thatofsplittingantichains,istreatedinChapterV.
The lectures primarily address basic extremal problems and inequalities – two
sidesofthesamecoin.Thus,theyalsopreparetowaysofthinkingandtomethods,
whichareusefulandapplicableinabroadermathematicalcontext.
viii Preface
Attheendofeverychapter,inadditiontoexercises,whichalsoopeneyesfornew
connections,wepresentseveralproblemsandsometimesweofferalsoconjectures.
Anotherimportantfeatureofthepresentbookisthatseveralofitsconceptsand
problems arise in response or by remodeling the given questions and the models
inscienceslikeInformationTheory,ComputerScience,orStatisticalPhysics.The
interdisciplinarycharactergivesArsCombinatoriaaspecialstatusinMathematics.
Again with occasional references to the books and reports we hope to create an
atmosphererichofincentivesfornewdiscoveries.
WehopethatArsCombinatoriagiveslightandjoytoallmindsandheartsstriving
for understanding and happiness through the world – from their origins to their
destinations.
Ournextremarksgotothepotentialreaders.Aboveall,asindicatedinthetitle,
thebookismeantforlecturesdealingwithadvancesinCombinatorics.
They are suited for all Mathematics students at the graduate level and for un-
dergraduatestudentswhoveryearlyspecializeincertainpartsofCombinatoricsto
combineit,forinstance,withComputerScience,especiallyComplexityTheoryor
DataStructures,orwithInformationTheory,especiallyCodingTheory,Computer
Systems Organization, and Communication Networks or Cryptology, or with Bio-
chemistry,especiallySequencinginGenetics,etc.
Chaptersofthebookcanbecombinedwiththebooksmentionedbelow.
ThereseemstobenobookonCombinatoricswithsuchaconcentrationofhigh-
level results, novel concepts, and advanced novel proof techniques. Therefore, the
bookisespeciallyrecommendedtoexperiencedresearchers.
Concerning content, there is an overlap with the research collection “Sperner
Theory” by K. Engel, Cambridge University Press, 1997, which, however, con-
tains no Combinatorial Number Theory. The two books “Combinatorics (Set
Systems, Hypergraphs, Families of Vectors, and Combinatorial Probability)”,
Cambridge University Press, 1986, by B. Bolloba´s and “A Course in Combina-
torics”,CambridgeUniversityPress,1992byJ.H.vanLintandR.M.Wilsontreat
basic combinatorial concepts and results. Probabilistic proofs can be learned from
“ProbabilisticMethodsinCombinatorics”byN.A.Alon,J.Spencer,andP.Erdo¨s,
AkademiaiKiado,Budapest,1974.
Allofthemarewell-suitedforundergraduateandgraduatecoursesandqualify
asexcellentpreparationsforchaptersofthismuchmoreadvancedbook,whichhas
as a special ingredient also an Appendix with a wealth of research problems and
conjecturessometimesconstitutingevenresearchprogramsforalreadyestablished
mathematicians,butalsoforPh.D.students.
There are also three research perspectives, novel links between Information
TheoryandCombinatorics:
• A direction in Extremal Theory of Sequences: Creating Order with Simple
Machines.
• InformationFlowsinNetworks.
• InformationTheoryandtheRegularityLemma.
Preface ix
Chapterscanalsobeusedtogivefreshairtograduatecourses.Perhapsthemost
recent and important are ChapterVI with results on divisors contributing to Ele-
mentary, Analytic, and Algebraic Number Theory, ChapterII with the Complete
IntersectionTheorem,whichrecentlyfoundconnectionstocomplexitytheory,and
ChapterIVwithsolutionstoextremalproblemswithdimensionconstraints,which
haveconsequencesforStatisticalDatabases.
For computer scientists, S. Jukna’s book “Extremal Combinatorics with Appli-
cationsinComputerScience”,Springer,2001,findsnowsubstantialadditionalma-
terialinmostchapters.
The senior author gratefully acknowledges that he was given the opportunity
to head two research projects “Combinatorics on Sequence Spaces” and “Models
with Information Exchange” in the Sonderforschungsbereich “Discrete Structures
inMathematics”oftheGermanScienceFoundations(DFG)inBielefeldfrom1989
to2000andaresearchproject“GeneralTheoryofInformationTransferandCom-
binatorics”attheCenterforInterdisciplinaryResearch(ZiF)inBielefeldfrom2000
to2004,andthatparallelandevenafterwardstheDFGhascontinueditsverygen-
eroussupportwithseveralprojects.
Most of the work reported here is an outgrowth of the cooperation with guests
in Bielefeld. Among them, essentially working in Combinatorics, were Noga
Alon,HaroutAydinian,ChristianBey,SergeiBezrukov, VladimirBlinovsky,Aart
Blokhuis, Ning Cai, Konrad Engel, Peter Erdo¨s, Levon Khachatrian, and Zhen
Zhang,andinNumberTheory,VladimirBlinovsky,PaulErdo¨s,ChristianMauduit,
LevonKhachatrian,andAndra´sSa´rko¨zy.
WeareindebtedtoChristianDeppeandChristianWischmannfordiligentproof
readingandthereductionofthenumberofRussianorGermanstyledphrases.
Lastbutnotleast,thejuniorauthorisgratefultoLeonidBassalygoforintrodu-
cinghimtoCombinatorialCodingTheoryintheearlyeightiesandtoDanKleitman
forinstructivediscussionsondiametricproblemsandtheseniorauthorisgratefulto
GyulaKatonaandDavidDaykinfortheirextremeencouragementintheseventies
tolookatExtremalSetTheory.
Table of Contents
ChapterI ConventionsandAuxiliaryResults 1
ChapterII IntersectionandDiametricProblems 9
Lecture1 TheCompleteIntersectionTheorem 9
Lecture2 TheDiametricProblemforVerticesintheHammingMetric 18
Lecture3 TheDiametricProblemforVerticesintheTaxiMetric 30
Lecture4 TheDiametricProblemforEdgesinHammingMetric 41
Lecture5 WordswithPairwiseCommonLetter 49
§1 AsymptoticalBehaviorofgn 50
q
Lecture6 ConstantDistanceCodePairs 52
§1 TheExactValueofM (n,δ) 52
q
§2 Four-WordsProperty 64
ChapterIII Covering,Packing,andListCodes 73
Lecture7 CoveringandPackingofHypergraphs 73
Lecture8 CoveringofProductsofGraphsandHypergraphs 83
Lecture9 MultiplePacking 89
§1 ALowerBound 91
§2 AnUpperBoundforR (ρ) 94
q,L
§3 AnUpperBoundforR (τ) 98
q,L
xi