The Goemans-Linial integrality gap for uniform sparsest cut is nearly square-root logarithmic

Theorem 1.1: there are an absolute $c>0$ and uniform-demand instances on $n_j\to\infty$ vertices with Goemans-Linial integrality gap at least $c\sqrt{\log n_j}/(\log\log n_j)^3$ . This improves the Kane-Meka bound $\exp(\Omega(\sqrt{\log\log n}))$ to within $(\log\log n)^3$ of the ARV upper bound. It is an existential sequence of sizes, not a bound at every $n$ ; it does not determine whether the gap is exactly $\Theta(\sqrt{\log n})$ , and it concerns the basic SDP, not Sherali-Adams or Lasserre strengthenings.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The Goemans-Linial integrality gap for uniform sparsest cut is nearly square-root logarithmic · Uniform sparsest cut SDP integrality gap

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VibeMathed record: The Goemans-Linial integrality gap for uniform sparsest cut is nearly square-root logarithmic

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the question; it gives $\mathrm{OPT}/\mathrm{GL}\ge c\sqrt{\log n}/(\log\log n)^3$ along a sequence $n\to\infty$ , which matches ARV up to a power of $\log\log n$ but does not settle the exact order. lean/formalization.yaml lists a main result for this manuscript (comparator UniformSparsestCut, declaration OAI.UniformSparsestCut.mainGap). The comparator statement was read here: it asserts an absolute $c>0$ and instances on $n_j\to\infty$ vertices with symmetric nonnegative capacities, positive Goemans-Linial value (negative-type distances realised in $\mathbb R^n$ with all triangle inequalities, normalised to total 1) and $c\sqrt{\log n_j}/(\log\log n_j)^3\le\mathrm{OPT}/\mathrm{GL}$ eventually, where OPT uses the denominator $|B|(n-|B|)$ . That is the headline claim. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.