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A history of central sets

Published online by Cambridge University Press:  04 June 2018

NEIL HINDMAN
NEIL HINDMAN*
Affiliation:
Department of Mathematics, Howard University, Washington, DC 20059, USA email nhindman@aol.com
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Abstract

We survey results about, and results using, central sets since their introduction in 1981.

Type
Survey Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 1 , January 2020 , pp. 1 - 33
Copyright
© Cambridge University Press, 2018 

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References

Adams, C., Hindman, N. and Strauss, D.. Largeness of the set of finite products in a semigroup. Semigroup Forum 76 (2008), 276–296.10.1007/s00233-007-9006-8Google Scholar
Barber, B., Hindman, N., Leader, I. and Strauss, D.. Distinguishing subgroups of the rationals by their Ramsey properties. J. Combin. Theory Ser. A 129 (2014), 93–104.10.1016/j.jcta.2014.10.002Google Scholar
Barber, B., Hindman, N., Leader, I. and Strauss, D.. Partition regularity without the columns property. Proc. Amer. Math. Soc. 143 (2015), 3387–3399.10.1090/S0002-9939-2015-12519-1Google Scholar
Barge, M. and Zamboni, L.. Central sets and substitutive dynamical systems. Adv. Math. 248 (2013), 308–323.10.1016/j.aim.2013.08.003Google Scholar
Beiglböck, M.. A multidimensional central sets theorem. Combin. Probab. Comput. 15 (2006), 807–814.10.1017/S0963548306007826Google Scholar
Beiglböck, M., Bergelson, V., Downarowicz, T. and Fish, A.. Solvability of Rado systems in D-sets. Topology Appl. 156 (2009), 2565–2571.10.1016/j.topol.2009.04.019Google Scholar
Beiglböck, M., Bergelson, V., Hindman, N. and Strauss, D.. Multiplicative structures in additively large sets. J. Combin. Theory Ser. A 113 (2006), 1219–1242.10.1016/j.jcta.2005.11.003Google Scholar
Beiglböck, M., Bergelson, V., Hindman, N. and Strauss, D.. Some new results in multiplicative and additive Ramsey theory. Trans. Amer. Math. Soc. 360 (2008), 819–847.10.1090/S0002-9947-07-04370-XGoogle Scholar
Bergelson, V.. Minimal idempotents and ergodic Ramsey theory. Topics in Dynamics and Ergodic Theory (London Mathematical Society Lecture Note Series, 310) . Cambridge University Press, Cambridge, 2003, pp. 8–39.10.1017/CBO9780511546716.004Google Scholar
Bergelson, V., Blass, A. and Hindman, N.. Partition theorems for spaces of variable words. Proc. Lond. Math. Soc. (3) 68 (1994), 449–476.Google Scholar
Bergelson, V. and Downarowicz, T.. Large sets of integers and hierarchy of mixing properties of measure preserving systems. Colloq. Math. 110 (2008), 117–150.Google Scholar
Bergelson, V. and Hindman, N.. Nonmetrizable topological dynamics and Ramsey theory. Trans. Amer. Math. Soc. 320 (1990), 293–320.Google Scholar
Bergelson, V. and Hindman, N.. Ramsey theory in non-commutative semigroups. Trans. Amer. Math. Soc. 330 (1992), 433–446.10.1090/S0002-9947-1992-1069744-5Google Scholar
Bergelson, V. and Hindman, N.. On IP*-sets and central sets. Combinatorica 14 (1994), 269–277.10.1007/BF01212975Google Scholar
Bergelson, V. and Hindman, N.. Partition regular structures contained in large sets are abundant. J. Combin. Theory Ser. A 93 (2001), 18–36.10.1006/jcta.2000.3061Google Scholar
Bergelson, V., Hindman, N. and Kra, B.. Iterated spectra of numbers—elementary, dynamical and algebraic approaches. Trans. Amer. Math. Soc. 348 (1996), 893–912.10.1090/S0002-9947-96-01533-4Google Scholar
Bergelson, V., Hindman, N. and Leader, I.. Additive and multiplicative Ramsey theory in the reals and the rationals. J. Combin. Theory Ser. A 85 (1999), 41–68.10.1006/jcta.1998.2892Google Scholar
Bergelson, V., Hindman, N. and Strauss, D.. Polynomials at iterated spectra near zero. Topology Appl. 158 (2011), 1815–1830.Google Scholar
Bergelson, V. and McCutcheon, R.. Central sets and a non-commutative Roth theorem. Amer. J. Math. 129 (2007), 1251–1275.10.1353/ajm.2007.0031Google Scholar
Bucci, M., Puzynina, S. and Zamboni, L.. Central sets generated by uniformly recurrent words. Ergod. Th. & Dynam. Sys. 35 (2015), 714–736.10.1017/etds.2013.69Google Scholar
Burns, S. and Hindman, N.. Quasi-central sets and their dynamical characterization. Topology Proc. 31 (2007), 445–455.Google Scholar
Carlson, T., Hindman, N., McLeod, J. and Strauss, D.. Almost disjoint large subsets of semigroups. Topology Appl. 155 (2008), 433–444.10.1016/j.topol.2005.05.012Google Scholar
Carlson, T., Hindman, N. and Strauss, D.. Ramsey theoretic consequences of some new results about algebra in the Stone–Čech compactification. Integers 5 (2005), A04, 1–26.Google Scholar
Carlson, T., Hindman, N. and Strauss, D.. An infinitary extension of the Graham–Rothschild parameter sets theorem. Trans. Amer. Math. Soc. 358 (2006), 3239–3262.10.1090/S0002-9947-06-03899-2Google Scholar
De, D. and Hindman, N.. Image partition regularity near zero. Discrete Math. 309 (2009), 3219–3232.10.1016/j.disc.2008.09.023Google Scholar
De, D., Hindman, N. and Strauss, D.. A new and stronger central sets theorem. Fund. Math. 199 (2008), 155–175.Google Scholar
De, D. and Paul, R.. Image partition regularity of matrices near 0 with real entries. New York J. Math. 17 (2011), 149–161.Google Scholar
Deuber, W.. Partitionen und lineare Gleichungssysteme. Math. Z. 133 (1973), 109–123.Google Scholar
Deuber, W. and Hindman, N.. Partitions and sums of (m, p, c)-sets. J. Combin. Theory Ser. A 45 (1987), 300–302.10.1016/0097-3165(87)90020-3Google Scholar
Farah, I., Hindman, N. and McLeod, J.. Partition theorems for layered partial semigroups. J. Combin. Theory Ser. A 98 (2002), 268–311.Google Scholar
Furstenberg, H.. Recurrence in Ergodic Theory and Combinatorical Number Theory. Princeton University Press, Princeton, NJ, 1981.10.1515/9781400855162Google Scholar
Furstenberg, H. and Katznelson, Y.. An ergodic Szemerédi theorem for IP-systems and combinatorial theory. J. Anal. Math. 45 (1985), 117–168.Google Scholar
Furstenberg, H. and Katznelson, Y.. Idempotents in compact semigroups and Ramsey theory. Israel J. Math. 68 (1989), 257–270.10.1007/BF02764984Google Scholar
Furstenberg, H. and Weiss, B.. Topological dynamics and combinatorial number theory. J. Anal. Math. 34 (1978), 61–85.10.1007/BF02790008Google Scholar
Glasner, S.. Divisibility properties and the Stone–Čech compactification. Canad. J. Math. 32 (1980), 993–1007.Google Scholar
Graham, R., Lin, S. and Lin, C.. Spectra of numbers. Math. Mag. 51 (1978), 174–176.10.1080/0025570X.1978.11976703Google Scholar
Graham, R. and Rothschild, B.. Ramsey’s theorem for n-parameter sets. Trans. Amer. Math. Soc. 159 (1971), 257–292.Google Scholar
Hales, A. and Jewett, R.. Regularity and positional games. Trans. Amer. Math. Soc. 106 (1963), 222–229.Google Scholar
Hindman, N.. Image partition regularity over the reals. New York J. Math. 9 (2003), 79–91.Google Scholar
Hindman, N.. Small sets satisfying the central sets theorem. Combinatorial Number Theory. Walter de Gruyter, Berlin, 2009, pp. 57–63.Google Scholar
Hindman, N. and Johnson, J.. Images of C sets and related large sets under nonhomogeneous spectra. Integers 12A (2012), A2.Google Scholar
Hindman, N., Jones, L. and Peters, M.. Left large subsets of free semigroups and groups that are not right large. Semigroup Forum 90 (2015), 374–385.10.1007/s00233-014-9622-zGoogle Scholar
Hindman, N. and Leader, I.. Image partition regularity of matrices. Combin. Probab. Comput. 2 (1993), 437–463.10.1017/S0963548300000821Google Scholar
Hindman, N. and Leader, I.. The semigroup of ultrafilters near 0. Semigroup Forum 59 (1999), 33–55.Google Scholar
Hindman, N., Leader, I. and Strauss, D.. Infinite partition regular matrices—solutions in central sets. Trans. Amer. Math. Soc. 355 (2003), 1213–1235.10.1090/S0002-9947-02-03191-4Google Scholar
Hindman, N., Leader, I. and Strauss, D.. Image partition regular matrices—bounded solutions and preservation of largeness. Discrete Math. 242 (2002), 115–144.10.1016/S0012-365X(01)00276-XGoogle Scholar
Hindman, N., Leader, I. and Strauss, D.. Extensions of infinite partition regular systems. Electron. J. Combin. 22(2) (2015), #P2.29.Google Scholar
Hindman, N. and Lisan, A.. Points very close to the smallest ideal of 𝛽S . Semigroup Forum 49 (1994), 137–141.10.1007/BF02573479Google Scholar
Hindman, N., Maleki, A. and Strauss, D.. Central sets and their combinatorial characterization. J. Combin. Theory Ser. A 74 (1996), 188–208.10.1006/jcta.1996.0048Google Scholar
Hindman, N. and Strauss, D.. Infinite partition regular matrices, II—extending the finite results. Topology Proc. 25 (2000), 217–255.Google Scholar
Hindman, N. and Strauss, D.. A simple characterization of sets satisfying the central sets theorem. New York J. Math. 15 (2009), 405–413.Google Scholar
Hindman, N. and Strauss, D.. Cartesian products of sets satisfying the central sets theorem. Topology Proc. 35 (2010), 203–223.Google Scholar
Hindman, N. and Strauss, D.. Algebra in the Stone–Čech Compactification: Theory and Applications, 2nd edn. de Gruyter, Berlin, 2012.Google Scholar
Hindman, N. and Strauss, D.. Separating Milliken–Taylor systems in ℚ. J. Combinatorics 5 (2014), 305–333.10.4310/JOC.2014.v5.n3.a3Google Scholar
Hindman, N., Strauss, D. and Zelenyuk, Y.. Large rectangular semigroups in Stone–Čech compactifications. Trans. Amer. Math. Soc. 355 (2003), 2795–2812.10.1090/S0002-9947-03-03276-8Google Scholar
Hindman, N. and Woan, W.. Central sets in semigroups and partition regularity of systems of linear equations. Mathematika 40 (1993), 169–186.10.1112/S0025579300006963Google Scholar
Johnson, J.. A dynamical characterization of $C$ -sets. Preprint, 2011, arXiv:1112.0715.Google Scholar
Johnson, J.. A new and simpler noncommutative central sets theorem. Topology Appl. 189 (2015), 10–24.10.1016/j.topol.2015.03.006Google Scholar
Li, J.. Dynamical characterization of C-sets and its application. Fund. Math. 216 (2012), 259–286.10.4064/fm216-3-4Google Scholar
McLeod, J.. Central sets in commutative adequate partial semigroups. Topology Proc. 29 (2005), 567–576.Google Scholar
Milliken, K.. Ramsey’s theorem with sums or unions. J. Combin. Theory Ser. A 18 (1975), 276–290.10.1016/0097-3165(75)90039-4Google Scholar
Polya, G.. Untersuchungen über Lücken und Singularitaten von Potenzreihen. Math. Z. 29 (1929), 549–640.10.1007/BF01180553Google Scholar
Rado, R.. Studien zur Kombinatorik. Math. Z. 36 (1933), 242–280.Google Scholar
Shi, H. and Yang, H.. Nonmetrizable topological dynamical characterization of central sets. Fund. Math. 150 (1996), 1–9.10.4064/fm-150-1-1-9Google Scholar
Taylor, A.. A canonical partition relation for finite subsets of 𝜔. J. Combin. Theory Ser. A 21 (1976), 137–146.Google Scholar
Zelenyuk, Y.. Principal left ideals of 𝛽G may be both minimal and maximal. Bull. Lond. Math. Soc. 45 (2013), 613–617.10.1112/blms/bds127Google Scholar
Zelenyuk, Y.. Left maximal idempotents in G ∗ . Adv. Math. 262 (2014), 593–603.Google Scholar