Actions on semigroups and an infinitary Gowers–Hales–Jewett Ramsey theorem

Lupini, Martino

doi:10.1090/tran/7337

Actions on semigroups and an infinitary Gowers–Hales–Jewett Ramsey theorem
HTML articles powered by AMS MathViewer

by Martino Lupini PDF

Trans. Amer. Math. Soc. 371 (2019), 3083-3116 Request permission

Abstract:

We introduce the notion of (Ramsey) action on a (filtered) semigroup. We then prove in this setting a general result providing a common generalization of the infinitary Gowers Ramsey theorem for multiple tetris operations, the infinitary Hales–Jewett theorems (for both located and nonlocated words), and the Farah–Hindman–McLeod Ramsey theorem for layered actions on partial semigroups. We also establish a polynomial version of our main result, recovering the polynomial Milliken–Taylor theorem of Bergelson–Hindman–Williams as a particular case. We present applications of our Ramsey-theoretic results to the structure of recurrence sets in amenable groups.

References

Ethan Akin, Recurrence in topological dynamics, The University Series in Mathematics, Plenum Press, New York, 1997. Furstenberg families and Ellis actions. MR 1467479, DOI 10.1007/978-1-4757-2668-8
Dana Bartošová and Aleksandra Kwiatkowska, Gowers’ Ramsey theorem with multiple operations and dynamics of the homeomorphism group of the Lelek fan, J. Combin. Theory Ser. A 150 (2017), 108–136. MR 3645569, DOI 10.1016/j.jcta.2017.02.002
Mathias Beiglböck, Vitaly Bergelson, and Alexander Fish, Sumset phenomenon in countable amenable groups, Adv. Math. 223 (2010), no. 2, 416–432. MR 2565535, DOI 10.1016/j.aim.2009.08.009
Vitaly Bergelson, Andreas Blass, and Neil Hindman, Partition theorems for spaces of variable words, Proc. London Math. Soc. (3) 68 (1994), no. 3, 449–476. MR 1262304, DOI 10.1112/plms/s3-68.3.449
Vitaly Bergelson and Tomasz Downarowicz, Large sets of integers and hierarchy of mixing properties of measure preserving systems, Colloq. Math. 110 (2008), no. 1, 117–150. MR 2353901, DOI 10.4064/cm110-1-4
Vitaly Bergelson, Hillel Furstenberg, and Randall McCutcheon, IP-sets and polynomial recurrence, Ergodic Theory Dynam. Systems 16 (1996), no. 5, 963–974. MR 1417769, DOI 10.1017/S0143385700010130
Vitaly Bergelson, Neil Hindman, and Kendall Williams, Polynomial extensions of the Milliken-Taylor theorem, Trans. Amer. Math. Soc. 366 (2014), no. 11, 5727–5748. MR 3256182, DOI 10.1090/S0002-9947-2014-05958-8
B. Blackadar, Operator algebras, Encyclopaedia of Mathematical Sciences, vol. 122, Springer-Verlag, Berlin, 2006. Theory of $C^*$-algebras and von Neumann algebras; Operator Algebras and Non-commutative Geometry, III. MR 2188261, DOI 10.1007/3-540-28517-2
Robert Ellis, Lectures on topological dynamics, W. A. Benjamin, Inc., New York, 1969. MR 0267561
Ilijas Farah, Neil Hindman, and Jillian McLeod, Partition theorems for layered partial semigroups, J. Combin. Theory Ser. A 98 (2002), no. 2, 268–311. MR 1899627, DOI 10.1006/jcta.2001.3237
H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, N.J., 1981. M. B. Porter Lectures. MR 603625, DOI 10.1515/9781400855162
W. T. Gowers, Lipschitz functions on classical spaces, European J. Combin. 13 (1992), no. 3, 141–151. MR 1164759, DOI 10.1016/0195-6698(92)90020-Z
W. T. Gowers, Ramsey methods in Banach spaces, Handbook of the geometry of Banach spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1071–1097. MR 1999191, DOI 10.1016/S1874-5849(03)80031-1
A. W. Hales and R. I. Jewett, Regularity and positional games, Trans. Amer. Math. Soc. 106 (1963), 222–229. MR 143712, DOI 10.1090/S0002-9947-1963-0143712-1
Neil Hindman, Finite sums from sequences within cells of a partition of $N$, J. Combinatorial Theory Ser. A 17 (1974), 1–11. MR 349574, DOI 10.1016/0097-3165(74)90023-5
Neil Hindman and Dona Strauss, Algebra in the Stone-Čech compactification, De Gruyter Textbook, Walter de Gruyter & Co., Berlin, 2012. Theory and applications; Second revised and extended edition [of MR1642231]. MR 2893605
Vassilis Kanellopoulos, A proof of W. T. Gowers’ $c_0$ theorem, Proc. Amer. Math. Soc. 132 (2004), no. 11, 3231–3242. MR 2073297, DOI 10.1090/S0002-9939-04-07320-4
Martino Lupini, Gowers’ Ramsey theorem for generalized tetris operations, J. Combin. Theory Ser. A 149 (2017), 101–114. MR 3620731, DOI 10.1016/j.jcta.2017.02.001
Keith R. Milliken, Ramsey’s theorem with sums or unions, J. Combinatorial Theory Ser. A 18 (1975), 276–290. MR 373906, DOI 10.1016/0097-3165(75)90039-4
Diana Ojeda-Aristizabal, Finite forms of Gowers’ theorem on the oscillation stability of $C_0$, Combinatorica 37 (2017), no. 2, 143–155. MR 3638339, DOI 10.1007/s00493-015-3223-7
Christian Schell, Idempotent ultrafilters and polynomial recurrence, arXiv:0711.0484 (2007).
Sławomir Solecki, Monoid actions and ultrafilter methods in Ramsey theory, arXiv:1611.06600 (2016).
Alan D. Taylor, A canonical partition relation for finite subsets of $\omega$, J. Combinatorial Theory Ser. A 21 (1976), no. 2, 137–146. MR 424571, DOI 10.1016/0097-3165(76)90058-3
Stevo Todorcevic, Introduction to Ramsey spaces, Annals of Mathematics Studies, vol. 174, Princeton University Press, Princeton, NJ, 2010. MR 2603812, DOI 10.1515/9781400835409
Konstantinos Tyros, Primitive recursive bounds for the finite version of Gowers’ $c_0$ theorem, Mathematika 61 (2015), no. 3, 501–522. MR 3415638, DOI 10.1112/S0025579314000345