Infinite monochromatic sumsets for colourings of the reals

Abstract

N. Hindman, I. Leader and D. Strauss proved that it is consistent that there is a finite colouring of 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 R there is an infinite X ⊆ R so that c ↾ X + X is constant.

Monochromatic sumset without large cardinals

“…We also show the Ramsey statement above is true in ZFC when r = 2. This answers two questions from [8].…”

“…This can be easily inferred from the context. [5], [8]). For any r ≥ 2, define a sequence of finite strings of natural numbers s l : l ≤ r such that for each l ≤ r, |s l | = r + l and s i (k) = 2 if k < 2l 4 otherwise.…”

“…To finish the proof, we basically need similar arguments as Claim 2.9 and step 5 from [8]. We supply a proof for completeness.…”

“…For example, Hindman, Leader and Strauss [5] showed that if 2 ω < ℵ ω then there exists some r ∈ ω such that R → + (ℵ 0 ) r . On the other hand, Komjáth, Leader, Russell, Shelah, Soukup and Vidnyánszky [8] showed that relative to the existence of an ω 1 -Erdős cardinal, it is consistent that for any r ∈ ω, R → + (ℵ 0 ) r . These results are optimal in a sense as there exist the following restrictions:…”

“…Remark 1.2. The continuum in the model of [8] is an ℵ-fixed point, which is very large. Over a ground model of GCH, Theorem 1.1 suggests that the most natural way to eliminate the obstacles from cardinal arithmetic works since by a result of Hindman, Leader and Strauss [5], if R → + (ℵ 0 ) r for all r < ω, then 2 ω ≥ ℵ ω+1 .…”