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Erdős #949 · ramsey theory

Let S⊂RS\subset \mathbb{R} be a set containing no solutions to a+b=ca+b=c. Must there be a set A⊆R\SA\subseteq \mathbb{R}\backslash S of cardinality continuum such that A+A⊆R\SA+A\subseteq \mathbb{R}\backslash S?

Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

Let S⊂RS\subset \mathbb{R} be a set containing no solutions to a+b=ca+b=c. Must there be a set A⊆R\SA\subseteq \mathbb{R}\backslash S of cardinality continuum such that A+A⊆R\SA+A\subseteq \mathbb{R}\backslash S?

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Background

Erdős suggests that if the answer is no one could consider the variant where we assume that SS is Sidon (i.e. all a+ba+b with a,b∈Sa,b\in S are distinct, aside from the trivial coincidences).

In the comments Dillies gives a positive proof of this, found by AlphaProof: in other words, if S⊂RS\subset \mathbb{R} is a Sidon set then there exists A⊆R\SA\subseteq \mathbb{R}\backslash S of cardinality continuum such that A+A⊆R\SA+A\subseteq \mathbb{R}\backslash S.

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #949 link
  2. [2]Upstream Lean 4 statement for Erdős #949 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 949 link

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