Ryser's conjecture on covers of r-partite hypergraphs

An $r$ -partite $r$ -uniform hypergraph $H$ has disjoint vertex parts $V_1,\ldots,V_r$ and every edge has exactly one vertex in each part. Let $\tau(H)$ be its covering number (least size of a vertex set meeting every edge) and $\nu(H)$ its matching number. Ryser's conjecture asserts $\tau(H)\le(r-1)\nu(H)$ . For $r=2$ this is Konig's theorem; Aharoni proved $r=3$ ; in the intersecting case ( $\nu=1$ ) it was known through $r=5$ (Gyarfas, Tuza), and truncated projective planes show it would be tight whenever $r-1$ is a prime power. Does every $r$ -partite $r$ -uniform hypergraph satisfy $\tau(H)\le(r-1)\nu(H)$ ?

Result: Disproved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Construction
Field: Extremal hypergraph theory; covers and matchings
Posed by: Herbert J. Ryser (attributed); an equivalent formulation appears in J. R. Henderson's 1971 Caltech thesis
Year posed: 1971
Years open: 55y
Solved: 2026-09-23
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 50 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Ryser's conjecture is false, already for intersecting hypergraphs. September 23: for every sufficiently large prime $s\equiv2\pmod3$ and every sufficiently large odd $n$ (threshold depending on $s$ ), there is an intersecting $(s^n+1)$ -partite $(s^n+1)$ -uniform hypergraph with $\tau=s^n+1$ , against the conjectured $s^n$ . September 27: for every sufficiently large prime $q$ , an intersecting $(q+1)$ -partite $(q+1)$ -uniform hypergraph with $\tau=q+1$ and exactly $q+1$ nonisolated vertices in each part. Not given: an explicit rank or example, or anything for small $r$ (cases $r=4,5$ remain open in general).

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. Two manuscripts: the extension-field construction (September 23, 2026) and the balanced prime-order construction (September 27, 2026); the later paper says its proof is independent, and the earlier imports only three elementary lemmas from it.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of both manuscripts was read against Ryser's conjecture as posed. lean/formalization.yaml lists two main results for the September 23 paper: OAI.RyserCoveringCounterexample.exists_prime_eventually_counterexampleRank (one prime p > 5, p = 2 mod 3, with counterexamples at rank p^n + 1 for all large prime n) and OAI.RyserOdd.eventualOddFailures_and_infinite (every large such prime, every large odd n, infinitely many ranks). Both challenge statements were read: finite r-partite hypergraphs, intersecting, matching number 1, cover number r. The balanced companion's challenge (BalancedRyser, OAI.Balanced.main_result) is not in the formalization catalogue; its statement was read and matches that paper's theorem. None was rebuilt here. All thresholds are existential and not numerical, so no explicit counterexample is given.

Ryser's conjecture on covers of r-partite hypergraphs

What was actually shown

What the AI did

Verification

Sources

Discussion