Erdős Problem #1026

Full solution by Boris Alexeev, Stijn Cambie, Terence Tao and Lawrence Wu with AlphaEvolve, Aristotle, Gemini and GPT, formalized in Lean (8 Dec 2025). 2 AI contributions are recorded in all.

Problem
Erdős problem #1026 (combinatorics); current status: solved (Lean)
Credit
Boris Alexeev, Stijn Cambie, Terence Tao and Lawrence Wu with AlphaEvolve, Aristotle, Gemini and GPT · AI working with humans
Checked
Lean formalizationListed in Tao's wiki
Timeline
First AI contribution 2025-12-07 · wiki data as of 2026-06-30
Sources
Problem page (erdosproblems.com)Tao's AI-contributions wiki
SOLVEDPer Tao's AI-contributions wiki

The problem

Let x1,…,xnx_1,\ldots,x_n be a sequence of distinct real numbers. Determine

max⁡(∑xir),\max\left(\sum x_{i_r}\right),

where the maximum is taken over all monotonic subsequences.

Statement quoted from erdosproblems.com/1026, maintained by Thomas Bloom. Original source: [Er71,p.107]. Current status there: solved (Lean).

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 Literature
7 Dec, 2025 AI on its own; related literature found afterwards Aristotle — 🟢 Full solution (Lean) 🟢 Tidor, Wang, and Yang (2016)
8 Dec, 2025 AI working with humans AlphaEvolve, Aristotle, Gemini, GPT Boris Alexeev, Stijn Cambie, Terence Tao, Lawrence Wu 🟢 Full solution (Lean) —

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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