The Simonovits-Sós conjecture on 3AP-intersecting families
A family of subsets of is 3AP-intersecting if any two members meet in a set containing a non-trivial three-term arithmetic progression. Simonovits and Sós conjectured that the largest such family has size , attained by fixing a progression. Before this work nothing better than the trivial bound was known. What is the maximum size?
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI co-developed
- Method
- Argument
- Field
- Extremal set theory
- Posed by
- Miklós Simonovits and Vera T. Sós, by personal communication to Chung, Graham, Frankl and Shearer, who recorded it in their 1986 paper
- Year posed
- 1986
- Years open
- 40y
- Solved
- 2026-09-16
- Model
- GPT-6 Astra
- Vendor
- OpenAI
- Collaborators
- Peter Keevash
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 25 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Partial: any 3AP-intersecting family has size at most for an absolute constant , the first bound below the trivial one. The conjectured maximum is not established. The same bound is proved for -intersecting families whenever is a 3-graph on with bounded codegrees, and a clique shows the codegree assumption cannot be removed.
What the AI did
From the paper's statement on AI use: the proof was found by GPT-6 Astra following a hint by the author, who then simplified and rewrote it.
Verification
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 for an absolute , generalised to -intersecting families for any 3-graph 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.