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Nonstandard methods in Ramsey theory and combinatorial number theory. (English) Zbl 1432.05001

Lecture Notes in Mathematics 2239. Cham: Springer (ISBN 978-3-030-17955-7/pbk; 978-3-030-17956-4/ebook). xvi, 204 p. (2019).
This book is organized into four parts – Part I (Preliminaries, Chapters 1–5); Part II (Ramsey theory, Chapters 6–9); Part III (Combinatorial number theory, Chapters 10–15); Part IV (Other topics, Chapters 16–17); and an appendix A (Foundations of nonstandard analysis).
Chapter 1 is devoted to basic introductions and properties of ultrafilters, which were introduced by H. Cartan [C. R. Acad. Sci., Paris 205, 777–779 (1937; Zbl 0018.00302; JFM 63.0569.03); ibid. 205, 595–598 (1937; Zbl 0017.24305)]. Ultrafilters and the corresponding construction of ultraproduct are common tools in mathematical logic, but they also found many applications in other areas of mathematics particularly in algebra, topology and functional analysis. The extension of the operation on a semigroup to the space of ultrafilters can be seen as a particular instance of the notion of Arens product on the bidual of a Banach algebra [R. Arens, Proc. Am. Math. Soc. 2, 839–848 (1951; Zbl 0044.32601)]. This was the initial approach taken in the study of the Stone-Čech compactification since 1950s [P. Civin and B. Yood, Pac. J. Math. 11, 847–870 (1961; Zbl 0119.10903); M. M. Day, Ill. J. Math. 1, 509–544 (1957; Zbl 0078.29402)]. Its realization as a space of ultrafilters was explicitly considered by R. Ellis [Lectures on topological dynamics. New York: W. A. Benjamin, Inc. (1969; Zbl 0193.51502)].
Chapter 2 is concerned to nonstandard analysis which was introduced by A. Robinson [Non-standard analysis. Amsterdam: North-Holland Publishing Company (1966; Zbl 0151.00803)], and his original approach was based on model theory. The authors discuss properties of the nonstandard versions of the natural, integer, rational, and real numbers; which is named by adding the prefix “hyper”. The fundamental principle of the star map is introduced. Even at this stage the treatment is still informal, and it is sufficient for the readers to gain first hand idea how nonstandard methods can be used in applications. After Robinson, W. A. J. Luxemburg [Non-standard analysis. Lectures on A. Robinson’s theory of infinitesimals and infinitely large numbers. Reprinted from the second edition, 1964. Mathematics Department, California Institute of Technology, Pasadena, Calif., (1973); Am. Math. Mon. 80, 38–67 (1973; Zbl 0268.26019)], which helped for further popularize nonstandard methods. The ultrapower approach is followed by R. Goldblatt [Lectures on the hyperreals. An introduction to nonstandard analysis. New York, NY: Springer (1998; Zbl 0911.03032)] proposed an alternative approach based on the ultrapower construction, which is an accessible introduction to nonstandard analysis. A survey of several different possible approaches to nonstandard methods is proposed by V. Benci et al. [Lect. Notes Log. 25, 3–44 (2006; Zbl 1104.03061)]. A nice introduction (with many examples) to nonstandard methods for number theorists is given by R. Jin [Bull. Symb. Log. 6, No. 3, 331–341 (2000; Zbl 1071.11502); Integers 8, No. 2, Article A07, 30 p. (2008; Zbl 1185.11011)].
In Chapter 3, the authors discuss strong connections between ultrafilters (as introduced in Chapter 1) and nonstandard analysis (as introduced in Chapter 2). The number theorists will recognize hyperfinite generators of ultrafilters simply as realizations of the types corresponding to the ultrafilters. A detailed survey on hyperfinite generator of ultrafilters and their properties is presented by M. Di Nasso [in: P. A. Loeb (ed.) and M. P. H. Wolff (ed.), Nonstandard analysis for the working mathematician. 2nd edition. Dordrecht: Springer (2015; Zbl 1357.03003), 443–474].
Chapter 4 is devoted to many stars iterated nonstandard extensions. The authors describe operators between ultrafilters, such as the Fubini product, interms of corresponding nonstandard points. In nonstandard method, one has to assume that the star map goes from the usual “standard” universe to a different (larger) “nonstandard” universe. The authors explain how one can dispense of this distinction assume that there is just one universe, which is mapped to itself by the star map. This yields the notion of iterated nonstandard extensions, which will be crucial in interpreting the Fubini product and other ultrafilter operators on the corresponding nonstandard points. Iterated hyper-extensions and their use to characterise sums of ultrafilters were introduced by Di Nasso in his unpublished lecture notes (2010). Further, this technique and its foundations was systematically studied by L. Luperi Baglini [Hyperintegers and nonstandard techniques in combinatorics of numbers. University of Siena (Ph.D. Thesis) (2012), arXiv:1212.2049]. He has also found several applications in combinatorial number theory. A use of iterated hyper-extensions for a nonstandard proof of Rado’s theorem is discussed by M. Di Nasso [Proc. Am. Math. Soc. 143, No. 4, 1749–1761 (2015; Zbl 1386.03057)].
Chapter 5 is devoted to Leob measure; the authors define this term and also explained about Lebesgue measure via this new concept. The Leob measure’s construction was introduced by P. A. Loeb [Trans. Am. Math. Soc. 211, 113–122 (1975; Zbl 0312.28004)]. The ergodic theory of hypercycle systems and its applications are discussed in brief. The proof of the general ergodic theorem is fairly non trivial; however the proof for the hypercycle system given by T. Kamae [Isr. J. Math. 42, 284–290 (1982; Zbl 0499.28011)] is also discussed.
Chapter 6 deals with Ramsey’s theorem, which was studied by F. P. Ramsey [Proc. Lond. Math. Soc. (2) 30, 264–286 (1929; JFM 55.0032.04)]. Ramsey’s theorem was rediscovered in 1950’s by Erdős and Rado, who recognized its fundamental importance and also provided several variations and applications including R. Rado’s decomposition theorem [Ann. Discrete Math. 3, 191–194 (1978; Zbl 0388.05031)].
Chapter 7 is devoted to the theorem of van der Waerden and Halls-Jewett. Please note that B. L. van der Waerden’s theorem [Nieuw Arch. Wiskd., II. Ser. 15, 212–216 (1927; JFM 53.0073.12)] is chronologically one of the first results in Ramsey’s theory, although preceded by the Hindman cube lemma in the paper of D. Hilbert [J. Reine Angew. Math. 110, 104–129 (1892; JFM 24.0087.03)] and by I. Schur’s lemma [Jahresber. Dtsch. Math.-Ver. 25, 114–117 (1916; JFM 46.0193.02)]. A. W. Hales and R. I. Jewett’s theorem [Trans. Am. Math. Soc. 106, 222–229 (1963; Zbl 0113.14802)] is an abstract Ramsey theoretic result motivated by the mathematical study of positional games such as “Tic-Tac-Toe” or “Go-Moku”. The original proof of Halls and Jewett [loc. cit.] was finitary and purely combinatorial. Also an infinitary proof was given by V. Bergelson et al. [Proc. Lond. Math. Soc. (3) 68, No. 3, 449–476 (1994; Zbl 0809.04005)]. R. L. Graham and B. L. Rothschild’s theorem [Trans. Am. Math. Soc. 159, 257–292 (1971; Zbl 0233.05003)] was motivated by a conjecture of Rota on a geometric analogue of Ramsey’s theorem. Rota’s conjecture was established by R. L. Graham et al. [Adv. Math. 8, 417–433 (1972; Zbl 0243.18011)].
Chapter 8 deals with mathematical devlopments from Hindman to Gowers. A conjecture introduced by Graham and Rothschild’s theorem [loc. cit.] was first proved by N. Hindman [J. Comb. Theory, Ser. A 17, 1–11 (1974; Zbl 0285.05012)] applying purely combinatorial methods is known as Hindman’s theorem on finite sums. Hindman’s theorem is another pigeonhole principle, which considers combinatorial configurations provided by sets of finite sums of infinite sequences. The authors also discuss about the Milliken-Taylor theorem and Folkman’s theorem; which is a straightforward consequence of van der Waerden’s theorem. However, the Milliken-Taylor theorem is a simultaneous generalization of Ramsey’s theorem (by taking finite sets \(F_1, \dots, F_m\), to have cardinality one) and Hindman’s theorem (by taking \(m=1\)). W. T. Gowers’ theorem [Eur. J. Comb. 13, No. 3, 141–151 (1992; Zbl 0763.46015)] was motivated by a problem on the geometry of the Banach space \(C_0\). Although Gowers’ original proof was infinitary and used ultrafilter methods, explicit purely combinatorial proofs of the corresponding finitary statement were later obtained by D. Ojeda-Aristizabal [Combinatorica 37, No. 2, 143–155 (2017; Zbl 1399.05221)] and K. Tyros [Mathematika 61, No. 3, 501–522 (2015; Zbl 1409.05208)]. The more general version of Gower’s theorem, which was already established by M. Lupini [J. Comb. Theory, Ser. A 149, 101–114 (2017; Zbl 1358.05299)] is presented in this chapter.
Chapter 9 is related to partition regularity of equations, the authors explain that how nonstandard methods can be used to prove the partition regularity of some Diophantine equations as well as how thay can be used to establish that some Diophantine equations are not partition regular. The classical theorem of Rado for a single equation using characterization of partition regularity is proved. Some valuable results concerning nonlinear Diophantine equations investigated by N. Hindman [Integers 11, No. 4, 431–439, A10 (2011; Zbl 1243.05237)], M. Di Nasso and M. Riggio [Combinatorica 38, No. 5, 1067–1078 (2018; Zbl 1438.11075)], P. Csikvári et al. [Combinatorica 32, No. 4, 425–449 (2012; Zbl 1286.11026)] and B. Green and T. Sanders [Discrete Anal. 2016, Paper No. 5, 43 p. (2016; Zbl 1400.11024)] are also discussed.
Chapter 10 is devoted to densities and structural properties, the authors introduce upper density and the Banach density as well as some structural notions of largeness for sets of natural numbers. The first appearance of nonstandard methods in connection with densities and structural properties seems to be S. C. Leth’s dissertation and his subsequent article [Stud. Log. 47, No. 3, 265–278 (1989; Zbl 0675.03042)]. The partition regularity of pricewise syndeticity was first proved by T. C. Brown [Pac. J. Math. 36, 285–289 (1971; Zbl 0211.33504)]. After 6 years, H. Furstenberg [J. Anal. Math. 31, 204–256 (1977; Zbl 0347.28016)] introduced his ergodic-theoretic proof of Szemerédi’s theorem. It was surely known to many experts that one could use hypercycle systems to prove the Furstenberg correspondence principle; but the first appearance of this idea in the literature seems to be generalizations of the Furstenberg correspondance due to H. Towsner [Ergodic Theory Dyn. Syst. 29, No. 4, 1309–1326 (2009; Zbl 1179.37006)].
Chapter 11 is about working in the remote realm. The authors discuss in brief finite embeddability between sets of natural numbers, Banach density as Shnirelmann density in the remote realm and their possible applications. The notion of finite embeddability was isolated and studied by M. Di Nasso [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)], although it was implicit in several previous papers of additive number theory. The idea of extending the finite embeddability relation to ultrafilters is due to L. Luperi Baglini [Arch. Math. Logic 55, No. 5–6, 705–734 (2016; Zbl 1354.05139)]; also it was jointly studied by A. Blass and M. Di Nasso [Bull. Pol. Acad. Sci., Math. 63, No. 3, 195–206 (2015; Zbl 1382.03068)]. R. Jin also made potential contribution on Banach density [J. Number Theory 91, No. 1, 20–38 (2001; Zbl 1071.11503)].
Chapter 12 is devoted to Jin’s sumset theorem, which is one of the earliest results in combinatorial number theory proven using nonstandard methods. The authors also discuss R. Jin’s original nonstandard proof [Proc. Am. Math. Soc. 130, No. 3, 855–861 (2002; Zbl 0985.03066)], as well as an ultrafilter proof given by M. Beiglböck [Isr. J. Math. 185, 369–374 (2011; Zbl 1300.11015)]; and also an alternative proof given by M. Di Nasso [Integers 14, Paper A27, 24 p. (2014; Zbl 1353.03084)]. Recent developments of quantitative strengthening of Jin’s sumset theorem are also discussed.
Chapter 13 is devoted to sumset configurations in sets of positive density. The authors discuss Erdős’ conjecture, which state that a set of natural numbers of positive lower density contains the sum of two infinite sets. Strauss provided a counter example to this conjecture [P. Erdős, Ann. Discrete Math. 6, 89–115 (1980; Zbl 0448.10002)], after that Erdős changed his conjecture; and now it is known as Erdős sumset conjecture [M. B. Nathanson, J. Comb. Theory, Ser. A 28, 150–155 (1980; Zbl 0451.10036)]. An I-shift version of Erdős’ conjecture; and a weak density version of Folkman’s theorem are also discussed.
Chapter 14 is concerned with near arithmetic progressions in sparse sets. The authors discuss a result of S. C. Leth [Proc. Am. Math. Soc. 134, No. 6, 1579–1589 (2006; Zbl 1090.11011)]; and its relation to the Erdős-Turán conjecture [P. Erdős and P. Turán, J. Lond. Math. Soc. 11, 261–264 (1936; Zbl 0015.15203; JFM 62.1126.01)].
Chapter 15 is related to the interval measure (IM) property. The authors define the notion due to S. C. Leth [Stud. Log. 47, No. 3, 265–278 (1989; Zbl 0675.03042)], which is related to internal subsets of the hypernatural numbers with the IM property. The notion of supra-SIM set due to I. Goldbring and S. Leth [“On supra-SIM sets of natural numbers”, Preprint, arXiv:1805.05933] is also discussed in brief.
Chapter 16 is related to triangle removal and Szemerédi regularity. The autCombinatorics. Combinatorial colloquium held at Keszthely, Hungary, from 28 June to 3 July 1976. Vol. I and II. Amsterdam-Oxford-New York: North-Holland Publishing Company. 939–945 (1978; Zbl 0393.05031)]. The valuable findings obtained by E. Szemerédi [Acta Arith. 27, 199–245 (1975; Zbl 0303.10056)], W. T. Gowers [Ann. Math. (2) 166, No. 3, 897–946 (2007; Zbl 1159.05052)], I. Goldbring and H. Towsner [Isr. J. Math. 199, Part B, 867–913 (2014; Zbl 1298.03100)] and T. Tao [J. Anal. Math. 103, 1–45 (2007; Zbl 1146.05038)] are also discussed.
Chapter 17 deals with approximate groups. There are many notion related to approximate groups appeared in the literature, the formal definition of an approximate group appearing in the chapter was introduced by T. Tao [Combinatorica 28, No. 5, 547–594 (2008; Zbl 1254.11017)]. The results presented in this chapter are similar to L. van den Dries [Astérisque 367–368, 79–113, Exp. No. 1077 (2015; Zbl 1358.11024)].

MSC:

05-02 Research exposition (monographs, survey articles) pertaining to combinatorics
05D10 Ramsey theory
05C55 Generalized Ramsey theory
11B75 Other combinatorial number theory
26E35 Nonstandard analysis
00-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematics in general
00-02 Research exposition (monographs, survey articles) pertaining to mathematics in general
03H05 Nonstandard models in mathematics
28E05 Nonstandard measure theory
30G06 Non-Archimedean function theory
46S20 Nonstandard functional analysis
47S20 Nonstandard operator theory
54J05 Nonstandard topology
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