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The Simonovits-Sós conjecture on 3AP-intersecting families

Partial: any 3AP-intersecting family has size at most $(\tfrac12 - c)2^n$ for an absolute constant $c>0$ , the first bound below the trivial one. The conjectured maximum $2^{n-3}$ is not established. The same bound is proved for $H$ -intersecting families whenever $H$ is a 3-graph on $[n]$ with bounded codegrees, and a clique shows the codegree assumption cannot be removed.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 16, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: preprint. AI contribution: ai-co-developed. 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 Simonovits-Sós conjecture on 3AP-intersecting families · 3AP-intersecting families

Confidence: Not scored

Registry verification: unreviewed · preprint · partial

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Attribution

VibeMathed
registry · event recorded by

GPT-6 Astra
model · ai model contributor · OpenAI

Peter Keevash
human · human collaborator

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Named evidence

VibeMathed record: The Simonovits-Sós conjecture on 3AP-intersecting families

specialist registry · passed

Checked here on 22 September 2026 against arXiv:2609.18870: the abstract calls this the first non-trivial progress towards the Simonovits-Sós conjecture and states the bound $(\tfrac12 - c)2^n$ for an absolute $c>0$ , generalised to $H$ -intersecting families for any 3-graph $H$ of bounded codegree, with a clique showing the codegree hypothesis cannot be dropped. The introduction attributes the conjecture to Simonovits and Sós by personal communication to the authors of reference [1], which is Chung, Graham, Frankl and Shearer, J. Combin. Theory Ser. A 43 (1986), so it dates to 1986 or earlier. Entered as Partial on the paper's own framing. The mathematics was not checked here; six days old, no referee.

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The Simonovits-Sós conjecture on 3AP-intersecting families

preprint · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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