Erdős Problem #120

Partial progress by DeepMind prover agent (14 Mar 2026): solution to variant problem (Lean).

Problem
Erdős problem #120 (combinatorics); current status: open
Credit
DeepMind prover agent · AI on its own; related literature found afterwards
Checked
Lean formalizationListed in Tao's wiki
Timeline
First AI contribution 2026-03-14 · wiki data as of 2026-06-30
Sources
Problem page (erdosproblems.com)Tao's AI-contributions wiki
PARTIALPer Tao's AI-contributions wiki

The problem

Let A⊆RA\subseteq\mathbb{R} be an infinite set. Must there be a set E⊂RE\subset \mathbb{R} of positive measure which does not contain any set of the shape aA+baA+b for some a,b∈Ra,b\in\mathbb{R} and a≠0a\neq 0?

Statement quoted from erdosproblems.com/120, maintained by Thomas Bloom. Original source: [Er74b][Er81b,p.29][Er83d][Er90][Er97f][Va99,2.46]. Current status there: open.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Date Kind AI systems Outcome Literature
14 Mar, 2026 AI on its own; related literature found afterwards DeepMind prover agent 🟡 Solution to variant problem (Lean) 🟡 de Reyna (1983)
14 Mar, 2026 Literature search GPT-5.4 Thinking 🟢 Solution to variant problem found —

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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