Claims

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2015_05_11_hindman_leader_strauss: Theorem 3.2 of Hindman, Leader and Strauss (Abh. Math. Semin. Univ. Hambg. 2017): a two-coloring of the reals under which no set of size continuum has monochromatic k-fold sums; under CH the case k = 2 answers the problem no.

2015_09_01_soukup_weiss: Corollary 3.2 of the unpublished Soukup--Weiss manuscript: in ZFC some two-coloring of the reals gives every uncountable set N-fold sums of both colors for every N at least 2, so the answer is no. Not refereed.

2016_01_01_komjath: Komjáth's theorem (Real Anal. Exchange 2016): in ZFC some two-coloring of the reals leaves no uncountable set with monochromatic pair sums, so the answer is no without CH. Refereed; not held here; a Lean file formalizes it.

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