Infinite monochromatic sumsets for colourings of the reals

Komjáth, Péter; Leader, Imre; Russell, Paul; Shelah, Saharon; Soukup, Dániel; Vidnyánszky, Zoltán

doi:10.1090/proc/14431

Infinite monochromatic sumsets for colourings of the reals
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by Péter Komjáth, Imre Leader, Paul A. Russell, Saharon Shelah, Dániel T. Soukup and Zoltán Vidnyánszky PDF

Proc. Amer. Math. Soc. 147 (2019), 2673-2684 Request permission

Abstract:

N. Hindman, I. Leader, and D. Strauss proved that it is consistent that there is a finite colouring of $\mathbb {R}$ so that no infinite sumset $X+X$ is monochromatic. Our aim in this paper is to prove a consistency result in the opposite direction: we show that, under certain set-theoretic assumptions, for any finite colouring $c$ of $\mathbb {R}$ there is an infinite $X\subseteq \mathbb {R}$ so that $c\upharpoonright X+X$ is constant.

References

Andreas Blass, Combinatorial cardinal characteristics of the continuum, Handbook of set theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 395–489. MR 2768685, DOI 10.1007/978-1-4020-5764-9_{7}
Lorenzo Carlucci, A note on Hindman-type theorems for uncountable cardinals, arXiv:1703.06706, 2017.
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
Shimon Garti and Saharon Shelah, Strong polarized relations for the continuum, Ann. Comb. 16 (2012), no. 2, 271–276. MR 2927607, DOI 10.1007/s00026-012-0130-0
Neil Hindman, Partitions and sums of integers with repetition, J. Combin. Theory Ser. A 27 (1979), no. 1, 19–32. MR 541340, DOI 10.1016/0097-3165(79)90004-9
Neil Hindman, Imre Leader, and Dona Strauss, Pairwise sums in colourings of the reals, Abh. Math. Semin. Univ. Hambg. 87 (2017), no. 2, 275–287. MR 3696151, DOI 10.1007/s12188-016-0166-x
Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR 1940513
Péter Komjáth, A certain 2-coloring of the reals, Real Anal. Exchange 41 (2016), no. 1, 227–231. MR 3511943
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
Imre Leader and Paul A. Russell, Monochromatic infinite sumsets, preprint, arXiv:1707.08071, 2017.
Donald Knuth, C. D. H. Cooper, J. C. Owings Jr., M. S. Klamkin, R. D. Whittekin, Jim King, Phil Hosford, and Michael Geraghty, Problems and Solutions: Elementary Problems: E2492-E2496, Amer. Math. Monthly 81 (1974), no. 8, 901–902. MR 1544067, DOI 10.2307/2319455
S. Shelah, Strong partition relations below the power set: consistency; was Sierpiński right? II, Sets, graphs and numbers (Budapest, 1991) Colloq. Math. Soc. János Bolyai, vol. 60, North-Holland, Amsterdam, 1992, pp. 637–668. MR 1218224
Dániel T. Soukup and William Weiss, Sums and anti-Ramsey colorings of $\mathbb {R}$, personal notes (http://www.logic.univie.ac.at/~soukupd73/finset_coloring.pdf).