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Steurer's Conjecture on Vectors with Small Average Correlation

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steurer-correlation-conjectureTheoretical computer scienceposed by David Steurer, 2010recorded: disproved

1 attempt · no person has looked

Statement

Steurer conjectured in 2010 that any family of nn unit vectors with polynomially small average correlation Ei,j∣⟨vi,vj⟩∣≤n−ε\mathbb{E}_{i,j}|\langle v_i,v_j\rangle| \le n^{-\varepsilon} contains linear-sized constant-separated sets. Refuted in a strong sense, using sparse high-dimensional expanders.

Context

A conjecture from the Sparsest Cut and unique games literature, where it was one route to understanding the integrality gap.

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  • #1

    Attempt 1

    constructionGPT-5.5 Pro via an agentic system on Codex with Farzam Ebrahimnejad, Shayan Oveis Gharan ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    GPT-5.5 Pro via an agentic system on Codex
    people
    Farzam Ebrahimnejad, Shayan Oveis Gharan

    Notable for the setup rather than the model: the authors used GPT-5.5 Pro primarily through an agentic system for long-horizon mathematical research, built by the first author on top of Codex and GPT models, and supplied it with their evolving research notes.

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