DISPROVED
This has been solved in the negative.
Is it true that, for any $2$-colouring of $\mathbb{R}$, there is a set $A\subseteq \mathbb{R}$ of cardinality $\aleph_1$ such that all sums $a+b$ with $a\neq b$ and $a,b\in A$ are the same colour?
In
[Er75b] Erdős reports that 'using the methods of our paper with Hajnal and Rado' he could prove that this is false assuming the continuum hypothesis.
A published proof of this is by Hindman, Leader, and Strauss
[HLS17] - in fact they prove, assuming the continuum hypothesis, that for any $k\geq 1$ there is a $2$-colouring of $\mathbb{R}$ such that for any uncountable $A\subseteq \mathbb{R}$ the set of $a_1+\cdots+a_k$ with $a_1,\ldots,a_k$ distinct elements of $A$ cannot be monochromatic.
This latter result (and hence a disproof of the original question) was proved without assuming the continuum hypothesis independently by Komjáth
[Ko16] and
Soukup and Weiss.
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This page was last edited 16 January 2026. View history
Additional thanks to: Kevin Barreto
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #965, https://www.erdosproblems.com/965, accessed 2026-04-11
