Result 162, Combinatorics

Counterexamples to Ryser’s covering conjecture

Disproves Ryser's covering conjecture by constructing intersecting $(q+1)$ -partite, $(q+1)$ -uniform hypergraphs with covering number $q+1$ , rather than the predicted bound q, for every sufficiently large prime q. A separate construction over extension fields also disproves Gyárfás's monochromatic tree-cover conjecture.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

Pairwise overlap in a family of sets might seem to make the whole family easy to hit with a few chosen elements. These manuscripts report counterexamples to a longstanding prediction about that relationship.

What changes?

A hypergraph is a family of sets called edges; a cover is a selection of vertices meeting every edge. The first manuscript reports, for every sufficiently large prime q, examples whose vertices form q+1 parts, with each edge taking one vertex from each part. Any two edges intersect, yet the smallest cover has q+1 vertices, exceeding Ryser's predicted bound of q. Each part has exactly q+1 nonisolated vertices, meaning vertices belonging to at least one edge.

What does that help mathematicians do?

The balanced examples show that equal numbers of participating vertices in each part do not rescue the proposed bound. The second manuscript reports the same one-vertex excess for intersecting hypergraphs with s^n+1 parts and s^n+1 vertices per edge, for every sufficiently large prime s congruent to 2 modulo 3 and every sufficiently large odd n, with the threshold for n depending on s. Researchers therefore need stronger assumptions to recover the proposed covering bound.

Are there practical applications?

The immediate value is foundational: identifying limits of covering principles in combinatorics. The supplied summary also reports a separate extension-field construction contradicting Gyárfás's monochromatic tree-cover conjecture, concerning covers of edge-colored graphs by single-color trees. This extends the claimed consequences to another covering problem, rather than establishing a practical algorithm or performance improvement.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Balanced counterexamples to Ryser's conjecture at prime orders

September 27, 2026 40 pages

For every sufficiently large prime q, we construct a finite intersecting $(q+1)$ -partite $(q+1)$ -uniform hypergraph with covering number $q+1$ and exactly $q+1$ nonisolated vertices in each part. This disproves Ryser's covering conjecture, even for intersecting hypergraphs with equally sized parts.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Balanced-Counterexamples-to-Rysers-Conjecture-at-Prime-Orders-September-27-2026,
  author = {{OpenAI}},
  title = {{Balanced counterexamples to Ryser's conjecture at prime orders}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Balanced-Counterexamples-to-Rysers-Conjecture-at-Prime-Orders-September-27-2026/paper.pdf}{OAI:Balanced-Counterexamples-to-Rysers-Conjecture-at-Prime-Orders-September-27-2026}},
  year = {2026}
}

A counterexample to Ryser's covering conjecture

September 23, 2026 39 pages Main result formalized in Lean

For every sufficiently large prime $s\equiv2\pmod3$ and every sufficiently large odd integer n, with the threshold depending on s, we construct an intersecting $(s^n+1)$ -partite $(s^n+1)$ -uniform hypergraph with covering number $s^n+1$ . This disproves Ryser's covering conjecture in its intersecting case.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026,
  author = {{OpenAI}},
  title = {{A counterexample to Ryser's covering conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/paper.pdf}{OAI:A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/162.md.

Counterexamples to Ryser’s covering conjecture

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

Ryser's covering conjecture predicts that an intersecting $r$ -partite hypergraph has a vertex cover of size at most $r-1$ . The formalization proves that every sufficiently large prime $q$ has a finite intersecting $(q+1)$ -partite, $(q+1)$ -uniform hypergraph with covering number $q+1$ and exactly $q+1$ nonisolated vertices in each part. Thus the conjecture fails even with equal part sizes. The prime threshold is existential.

Ryser's covering conjecture predicts $\tau\le(r-1)\nu$ for an $r$ -partite hypergraph, where $\tau$ and $\nu$ are its covering and matching numbers. The formalized constructions give finite intersecting $r$ -partite $r$ -uniform hypergraphs with $\nu=1$ and $\tau=r$ , contradicting the bound. The ranks have the form $r=p^n+1$ : one fixed prime $p\equiv2\pmod3$ works for all sufficiently large prime degrees $n$ , and a second result covers every sufficiently large such prime $p$ and every sufficiently large odd $n$ , with the degree threshold allowed to depend on $p$ . Infinitely many ranks occur. Equal part sizes and numerical thresholds are not asserted.

Comparator links

Result	Comparator statement
Balanced Ryser counterexamples at prime orders	BalancedRyser.lean
Fixed-prime Ryser counterexamples	RyserCovering.lean
Counterexamples in odd extension degrees	RyserOddExtensions.lean

At a glance

Subject: Combinatorics
Manuscripts: 2
Pages: 79
Dated: Sep 23 to Sep 27, 2026
Lean formalization: Yes
Reasoning summary: No

Named problems addressed

Ryser's covering conjectureDisproved
Gyárfás's monochromatic tree-cover conjectureDisproved

Indexed by Emergent Mind with an LLM from the text above; check the manuscripts for the precise statements.

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