Erdős Problem #52

The problem

Let $A$ be a finite set of integers. Is it true that for every $\epsilon>0$
$\max( \lvert A+A\rvert,\lvert AA\rvert)\gg_\epsilon \lvert A\rvert^{2-\epsilon}?$

Statement quoted from erdosproblems.com/52, maintained by Thomas Bloom. Original source: [Er77c][Er80,p.112][ErGr80][Er91][Er92c][Er95][Er97][Er97e][Va99,1.26]. 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	Humans	Outcome
28 May, 2026	AI working with humans	GPT-5.5 Pro	Thomas Bloom, Will Sawin, Carl Schildkraut, Dmitrii Zhelezov	🟡 Solution to variant problem
28 May, 2026	AI working with humans	GPT-5.5	Ingo Althöfer	🟡 Improved bound to variant problem
3 Jun, 2026	AI on its own	Claude Mythos	—	🟡 Solution to variant problem
4 Jun, 2026	AI working with humans	GPT-5.5	Boris Alexeev	⚪ Candidate solution to variant problem

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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