Hindman-like theorems with uncountably many colours and finite monochromatic sets

Fernández-Bretón, David; Lee, Sung

doi:10.1090/proc/14649

Hindman-like theorems with uncountably many colours and finite monochromatic sets
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by David Fernández-Bretón and Sung Hyup Lee PDF

Proc. Amer. Math. Soc. 148 (2020), 3099-3112 Request permission

Abstract:

A particular case of the Hindman–Galvin–Glazer theorem states that, for every partition of an infinite abelian group $G$ into two cells, there will be an infinite $X\subseteq G$ such that the set of its finite sums $\{x_1+\cdots +x_n \mid n\in \mathbb N\wedge x_1,\ldots ,x_n\in X\text { are distinct}\}$ is monochromatic. It is known that the same statement is false, in a very strong sense, if one attempts to obtain an uncountable (rather than just infinite) $X$. On the other hand, a recent result of Komjáth states that, for partitions into uncountably many cells, it is possible to obtain monochromatic sets of the form $\mathrm {FS}(X)$, for $X$ of some prescribed finite size, when working with sufficiently large Boolean groups. In this paper, we provide a generalization of Komjáth’s result, and we show that, in a sense, this generalization is the strongest possible.

References

Lorenzo Carlucci, A note on Hindman-type theorems for uncountable cardinals, Order 36 (2019), no. 1, 19–22. MR 3925577, DOI 10.1007/s11083-018-9452-9
G. Elekes, A. Hajnal, and P. Komjáth, Partition theorems for the power set, Sets, graphs and numbers (Budapest, 1991) Colloq. Math. Soc. János Bolyai, vol. 60, North-Holland, Amsterdam, 1992, pp. 211–217. MR 1218191
David J. Fernández Bretón, Every strongly summable ultrafilter on $\bigoplus \Bbb Z_2$ is sparse, New York J. Math. 19 (2013), 117–129. MR 3065919
David J. Fernández-Bretón, Hindman’s theorem is only a countable phenomenon, Order 35 (2018), no. 1, 83–91. MR 3774507, DOI 10.1007/s11083-016-9419-7
David Fernández-Bretón and Assaf Rinot, Strong failures of higher analogs of Hindman’s theorem, Trans. Amer. Math. Soc. 369 (2017), no. 12, 8939–8966. MR 3710649, DOI 10.1090/tran/7131
R. L. Graham and B. L. Rothschild, Ramsey’s theorem for $n$-parameter sets, Trans. Amer. Math. Soc. 159 (1971), 257–292. MR 284352, DOI 10.1090/S0002-9947-1971-0284352-8
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
Péter Komjáth, Note: a Ramsey statement for infinite groups, Combinatorica 38 (2018), no. 4, 1017–1020. MR 3850014, DOI 10.1007/s00493-016-3706-1
Péter Komjáth, Set-theoretic constructions in Euclidean spaces, New trends in discrete and computational geometry, Algorithms Combin., vol. 10, Springer, Berlin, 1993, pp. 303–325. MR 1228048, DOI 10.1007/978-3-642-58043-7_{1}3
Péter Komjáth, Imre Leader, Paul A. Russell, Saharon Shelah, Dániel T. Soukup, and Zoltán Vidnyánszky, Infinite monochromatic sumsets for colourings of the reals, Proc. Amer. Math. Soc. 147 (2019), no. 6, 2673–2684. MR 3951442, DOI 10.1090/proc/14431
Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. An introduction to independence proofs. MR 597342
I. Leader and P. Russell, Monochromatic Infinite Sumsets, ArXiv preprint, ArXiv:1707.08071.
B. L. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Arch. Wiskd. 15 (1927), 212–216.
Yevhen Zelenyuk, Partitions of vector spaces over finite fields, Algebraic geometry and its applications, Ser. Number Theory Appl., vol. 5, World Sci. Publ., Hackensack, NJ, 2008, pp. 505–511. MR 2484073, DOI 10.1142/9789812793430_{0}028