Actions on semigroups and an infinitary Gowers-Hales-Jewett Ramsey theorem. (English) Zbl 1402.05212
Summary: 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.
MSC:
| 05D10 | Ramsey theory |
| 54D80 | Special constructions of topological spaces (spaces of ultrafilters, etc.) |
| 20M99 | Semigroups |
| 05C05 | Trees |
| 06A06 | Partial orders, general |
Keywords:
Gowers Ramsey theorem; Galvin-Glazer-Hindman theorem; Milliken-Taylor theorem; Hales-Jewett theorem; tetris operation; variable word; partial semigroup; ultrafilter; Stone-Čech compactificationReferences:
| [1] | Akin, Ethan, Recurrence in topological dynamics, The University Series in Mathematics, x+265 pp. (1997), Plenum Press, New York · Zbl 0919.54033 |
| [2] | Barto\v sov\'a, Dana; Kwiatkowska, Aleksandra, Gowers’ Ramsey theorem with multiple operations and dynamics of the homeomorphism group of the Lelek fan, J. Combin. Theory Ser. A, 150, 108-136 (2017) · Zbl 1377.05192 |
| [3] | Beiglb\"ock, Mathias; Bergelson, Vitaly; Fish, Alexander, Sumset phenomenon in countable amenable groups, Adv. Math., 223, 2, 416-432 (2010) · Zbl 1187.43002 |
| [4] | Bergelson, Vitaly; Blass, Andreas; Hindman, Neil, Partition theorems for spaces of variable words, Proc. London Math. Soc. (3), 68, 3, 449-476 (1994) · Zbl 0809.04005 |
| [5] | Bergelson, Vitaly; Downarowicz, Tomasz, Large sets of integers and hierarchy of mixing properties of measure preserving systems, Colloq. Math., 110, 1, 117-150 (2008) · Zbl 1142.37003 |
| [6] | Bergelson, Vitaly; Furstenberg, Hillel; McCutcheon, Randall, IP-sets and polynomial recurrence, Ergodic Theory Dynam. Systems, 16, 5, 963-974 (1996) · Zbl 0958.28013 |
| [7] | Bergelson, Vitaly; Hindman, Neil; Williams, Kendall, Polynomial extensions of the Milliken-Taylor theorem, Trans. Amer. Math. Soc., 366, 11, 5727-5748 (2014) · Zbl 1359.03036 |
| [8] | Blackadar, B., Operator algebras, Encyclopaedia of Mathematical Sciences, vol. 122, Operator Algebras and Non-commutative Geometry, III, xx+517 pp. (2006), Springer-Verlag, Berlin · Zbl 1092.46003 |
| [9] | Ellis, Robert, Lectures on topological dynamics, xv+211 pp. (1969), W. A. Benjamin, Inc., New York · Zbl 0193.51502 |
| [10] | Farah, Ilijas; Hindman, Neil; McLeod, Jillian, Partition theorems for layered partial semigroups, J. Combin. Theory Ser. A, 98, 2, 268-311 (2002) · Zbl 0991.05100 |
| [11] | Furstenberg, H., Recurrence in ergodic theory and combinatorial number theory, M. B. Porter Lectures, xi+203 pp. (1981), Princeton University Press, Princeton, N.J. · Zbl 0459.28023 |
| [12] | Gowers, W. T., Lipschitz functions on classical spaces, European J. Combin., 13, 3, 141-151 (1992) · Zbl 0763.46015 |
| [13] | Gowers, W. T., Ramsey methods in Banach spaces. Handbook of the geometry of Banach spaces, Vol.2, 1071-1097 (2003), North-Holland, Amsterdam · Zbl 1043.46018 |
| [14] | Hales, A. W.; Jewett, R. I., Regularity and positional games, Trans. Amer. Math. Soc., 106, 222-229 (1963) · Zbl 0113.14802 |
| [15] | Hindman, Neil, Finite sums from sequences within cells of a partition of \(N\), J. Combinatorial Theory Ser. A, 17, 1-11 (1974) · Zbl 0285.05012 |
| [16] | Hindman, Neil; Strauss, Dona, Algebra in the Stone-\v Cech compactification, De Gruyter Textbook, xviii+591 pp. (2012), Walter de Gruyter & Co., Berlin · Zbl 1241.22001 |
| [17] | Kanellopoulos, Vassilis, A proof of W. T. Gowers’ \(c_0\) theorem, Proc. Amer. Math. Soc., 132, 11, 3231-3242 (2004) · Zbl 1058.46008 |
| [18] | Lupini, Martino, Gowers’ Ramsey theorem for generalized tetris operations, J. Combin. Theory Ser. A, 149, 101-114 (2017) · Zbl 1358.05299 |
| [19] | Milliken, Keith R., Ramsey’s theorem with sums or unions, J. Combinatorial Theory Ser. A, 18, 276-290 (1975) · Zbl 0323.05001 |
| [20] | Ojeda-Aristizabal, Diana, Finite forms of Gowers’ theorem on the oscillation stability of \(C_0\), Combinatorica, 37, 2, 143-155 (2017) · Zbl 1399.05221 · doi:10.1007/s00493-015-3223-7 |
| [21] | schnell_idempotent_2007 Christian Schell, Idempotent ultrafilters and polynomial recurrence, arXiv:0711.0484 (2007). |
| [22] | solecki_general_2016 Sawomir Solecki, Monoid actions and ultrafilter methods in Ramsey theory, arXiv:1611.06600 (2016). |
| [23] | Taylor, Alan D., A canonical partition relation for finite subsets of \(\omega \), J. Combinatorial Theory Ser. A, 21, 2, 137-146 (1976) · Zbl 0341.05010 |
| [24] | Todorcevic, Stevo, Introduction to Ramsey spaces, Annals of Mathematics Studies 174, viii+287 pp. (2010), Princeton University Press, Princeton, NJ · Zbl 1205.05001 |
| [25] | Tyros, Konstantinos, Primitive recursive bounds for the finite version of Gowers’ \(c_0\) theorem, Mathematika, 61, 3, 501-522 (2015) · Zbl 1409.05208 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
