Table Of ContentLecture Notes in Mathematics 2239
Mauro Di Nasso
Isaac Goldbring
Martino Lupini
Nonstandard Methods
in Ramsey Theory
and Combinatorial
Number Theory
Lecture Notes in Mathematics 2239
Editors-in-Chief:
Jean-MichelMorel,Cachan
BernardTeissier,Paris
AdvisoryEditors:
MichelBrion,Grenoble
CamilloDeLellis,Princeton
AlessioFigalli,Zurich
DavarKhoshnevisan,SaltLakeCity
IoannisKontoyiannis,Cambridge
GáborLugosi,Barcelona
MarkPodolskij,Aarhus
SylviaSerfaty,NewYork
AnnaWienhard,Heidelberg
Moreinformationaboutthisseriesathttp://www.springer.com/series/304
Mauro Di Nasso (cid:129) Isaac Goldbring (cid:129) Martino Lupini
Nonstandard Methods
in Ramsey Theory
and Combinatorial Number
Theory
123
MauroDiNasso IsaacGoldbring
DepartmentofMathematics DepartmentofMathematics
UniversitadiPisa UniversityofCalifornia,Irvine
Pisa,Italy Irvine,CA,USA
MartinoLupini
SchoolofMathematicsandStatistics
VictoriaUniversityofWellington
Wellington,NewZealand
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-3-030-17955-7 ISBN978-3-030-17956-4 (eBook)
https://doi.org/10.1007/978-3-030-17956-4
MathematicsSubjectClassification(2010):Primary:05D10,Secondary:03H10
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To Marian,Karina,and Nicola
Preface
Generallyspeaking,Ramsey theorystudieswhichcombinatorialconfigurationsof
a structure can always be found in one of the pieces of a given finite partition.
More generally, it considers the problem of which combinatorial configurations
can be found in sets that are “large” in some suitable sense. Dating back to the
foundationalresultsofvanderWaerden,Ramsey,Erdo˝s,Turán,andothersfromthe
1920sand1930s,Ramseytheoryhassincethenhadanextraordinarydevelopment.
On the one hand, many applications of Ramsey theory to numerous other areas
of mathematics, ranging from functional analysis, topology, and dynamics to set
theory,model theory,and computerscience, have been found.On the other hand,
resultsandmethodsfromotherareasofmathematicshavebeensuccessfullyapplied
to establish new results in Ramsey theory. For instance, ergodic theory has had a
profound impact on Ramsey theory, giving rise to the research area of “ergodic
Ramsey theory.” Perhaps the best known achievement of this approach is the
ergodic-theoretic proof of Szemerédi’s theorem due to Furstenberg in the 1980s.
Inadifferent(butintimatelyrelated)direction,thetheoryofultrafiltershasbeenan
importantsourceofmethodsandideasforRamseytheory.Inparticular,thestudyof
topologicalandalgebraicpropertiesofthespaceofultrafiltershasbeenusedtogive
shortandelegantproofsofdeepcombinatorialpigeonholeprinciples.Paradigmatic
inthisdirectionistheGalvin–GlazerultrafilterproofofHindman’stheoremonsets
of finite sums, previously established by Hindman in 1974 via a delicate, purely
combinatorialargument.
Recently, a new thread of research has emerged, where problems in Ramsey
theory are studied from the perspective of nonstandard analysis and nonstandard
methods. Developed by Abraham Robinson in the 1960s and based on first-order
logic and model theory, nonstandard analysis provided a formal and rigorous
treatment of calculus and classical analysis via infinitesimals. Such a treatment is
moresimilarinspirittotheapproachoriginallytakeninthedevelopmentofcalculus
in the seventeenth and eighteenth century and avoids the epsilon-delta arguments
thatareinherentinitslaterformalizationduetoWeierstrass.Whilethisisperhapsits
mostwell-knownapplication,nonstandardanalysisisactuallymuchmoreversatile.
Thefoundationsofnonstandardanalysisprovideuswithamethod,whichweshall
vii
viii Preface
callthenonstandardmethod,thatisapplicabletovirtuallyanyareaofmathematics.
Thenonstandardmethodhasthusfarbeenusedinnumerousareasofmathematics,
includingfunctionalanalysis,measuretheory,ergodictheory,differentialequations,
andstochasticanalysis,justtonameafewsuchareas.
In a nutshell, the nonstandard method allows one to extend the given math-
ematical universe and thus regard it as contained in a much richer nonstandard
universe. Such a nonstandarduniverse satisfies strong saturation properties which
in particular allow one to consider limiting objects which do not exist in the
standarduniverse.Thisprocedureissimilartopassingtoanultrapower,andinfact
the nonstandard method can also be seen as a way to axiomatize the ultrapower
construction in a way that distillates its essential features and benefits, but avoids
being bogged down by the irrelevant details of its concrete implementation. This
limiting process allows one to reformulate a given problem involving finite (but
arbitrarily large) structures or configurations into a problem involving a single
structure or configurationwhich is infinite but for all purposesbehaves as though
itwerefinite(intheprecisesensethatitishyperfiniteinthenonstandarduniverse).
Thisreformulationcanthenbetackleddirectlyusingfinitarymethods,rangingfrom
combinatorialcountingargumentstorecurrencetheoremsformeasurabledynamics,
recastinthenonstandarduniverse.
In the setting of Ramsey theory and combinatorics, the application of non-
standard methods was pioneered by the work of Keisler, Leth, and Jin from the
1980sand1990s.Theseapplicationsfocusedondensityproblemsincombinatorial
number theory. The general goal in this area is to establish the existence of
combinatorial configurations in sets that are large in the sense that they have
positive asymptotic density. For example, the aforementioned celebrated theorem
of Szemerédi from 1970 asserts that a set of integers of positive density contains
arbitrarily long finite arithmetic progressions. One of the contributions of the
nonstandardapproachistotranslatethenotionofasymptoticdensityontheintegers,
whichdoesnotsatisfyallthepropertiesofameasure,intoanactualmeasureinthe
nonstandard universe. This translation then makes methods from measure theory
and ergodic theory, such as the ergodic theorem or other recurrence theorems,
availableforthestudyofdensityproblems.Inasense,thiscanbeseenasaversion
of Furstenberg’scorrespondence(between sets of integers and measurable sets in
a dynamical system), with the extra feature that the dynamical system obtained
perfectly reflects all the combinatorial properties of the set that one started with.
The achievements of the nonstandard approach in this area include the work of
Leth on arithmetic progressions in sparse sets, Jin’s theorem on sumsets, as well
asJin’sFreiman-typeresultsoninverseproblemsforsumsets.Morerecently,these
methods have also been used by Jin, Leth, Mahlburg, and the present authors to
tackleaconjectureofErdo˝sconcerningsumsofinfinitesets(theso-calledB +C
conjecture),leadingtoitseventualsolutionbyMoreira,Richter,andRobertson.
Nonstandard methods are also tightly connected with ultrafilter methods. This
has been made precise and successfully appliedin a recent work of one of us (Di
Nasso),whereheobservedthatthereisaperfectcorrespondencebetweenultrafilters
andelementsofthenonstandarduniverseuptoanaturalnotionofequivalence.On
Preface ix
theonehand,thisallowsonetomanipulateultrafiltersasnonstandardpointsandto
useultrafiltermethodstoprovetheexistenceofcertaincombinatorialconfigurations
in the nonstandard universe. On the other hand, this gives an intuitive and direct
waytoinfer,fromtheexistenceofcertainultrafilterconfigurations,theexistenceof
correspondingstandardcombinatorialconfigurationsviathefundamentalprinciple
of transfer from nonstandard analysis. This perspective has successfully been
appliedbyDiNassoandLuperiBaglinitothestudyofpartitionregularityproblems
for Diophantineequationsoverthe integers, providingin particular a far-reaching
generalization of the classical theorem of Rado on partition regularity of systems
oflinearequations.UnlikeRado’stheorem,thisrecentgeneralizationalsoincludes
equationsthatarenotlinear.
Finally, it is worth mentioningthat manyother results in combinatoricscan be
seen,directlyorindirectly,asapplicationsofthenonstandardmethod.Forinstance,
thegroundbreakingworkofHrushovskiandBreuillard–Green–Taoonapproximate
groups,althoughnotoriginallypresentedin this way,admita naturalnonstandard
treatment. The same applies to the work of Bergelson and Tao on recurrence in
quasirandomgroups.
The goal of this manuscript is to introduce the uninitiated reader to the non-
standard method and to provide an overview of its most prominent applications
in Ramsey theory and combinatorial number theory. In particular, no previous
knowledge of nonstandard analysis will be assumed. Instead, we will provide a
complete and self-contained introduction to the nonstandard method in the first
part of this book. Novel to our introduction is a treatment of the topic of iterated
hyperextensions,which is crucialfor some applicationsand has thusfar appeared
onlyinspecializedresearcharticles.The intendedaudienceforthisbookincludes
researchers in combinatorics who desire to get acquainted with the nonstandard
approach,as wellaslogiciansandexpertsof nonstandardanalysiswhohavebeen
workinginthisorotherareasofresearch.Thelistofapplicationsofthenonstandard
method to combinatorics and Ramsey theory presented here is quite extensive,
includingcornerstone results of Ramsey theory such as Ramsey’s theorem, Hind-
man’stheoremonsetsoffinitesums,theHales–Jewetttheoremonvariablewords,
andGowers’Ramseytheorem.Itthenproceedswithresultsonpartitionregularityof
Diophantineequationsandwithdensityproblemsincombinatorialnumbertheory.
A nonstandardtreatment of the triangle removallemma, the Szemerédi regularity
lemma, and of the already mentionedwork of Hrushovskiand Breuillard–Green–
Tao on approximategroupsconcludethe book.We hope thatsuch a completelist
of examples will help the reader unfamiliar with the nonstandard method get a
goodgraspon how the approachworksand can be applied.At the same time, we
believe that collecting these results together, and providing a unified presentation
and approach,will provide a useful reference for researchersin the field and will
furtherstimulatetheresearchinthiscurrentlyveryactivearea.
Thisworkoriginatedoutoflecture noteswrittenbythe authorsonoccasionof
theworkshop“Nonstandardmethodsincombinatorialnumbertheory”heldin2017
atthe AmericanInstitute of Mathematics.We would like to thankthe participants
of the conference for providing valuable feedback to an earlier version of the
x Preface
manuscript. We received further useful comments and suggestions from several
people,includingKevinBarnum,JordanBarrett,RyanBurkhart,theBogotáLogic
Group(especiallyAlexBerensteinandDarioGarcia),andLorenzoLuperiBaglini.
WewouldalsoliketoexpressourgratitudetoourfriendandcolleagueStevenLeth
forhis manyhelpfulinsightson earlier versionsof this draft(as well as forbeing
responsibleforbringingthesecondauthorintothislineofresearch).
Di Nasso’s researchwas partiallysupportedbyPRIN Grant2012“Modelsand
Set Theory.” Goldbring’s work was partially supported by NSF CAREER grant
DMS-1349399. Lupini was partially supported by the NSF Grant DMS-1600186
andbyaResearchEstablishmentGrantbyVictoriaUniversityofWellington.
Pisa,Italy MauroDiNasso
Irvine,CA,USA IsaacGoldbring
Wellington,NewZealand MartinoLupini
