Result 173, Combinatorics

Seymour’s second-neighborhood conjecture

Proves Seymour's second-neighborhood conjecture: every nonempty finite oriented graph has a vertex with at least as many vertices at directed distance exactly two as at directed distance one. Oriented graphs may be arbitrary apart from the exclusion of loops and oppositely directed edge pairs.

Lean formalization Proof

The bigger picture

Why it matters

In any finite directed network without self-links or opposing links, the manuscript claims there is a starting point whose second layer of neighbors is at least as large as its first. This would establish a universal constraint on directed connectivity.

What changes?

The manuscript reports a proof of Seymour's second-neighborhood conjecture for every nonempty finite oriented graph, with no further restrictions. Such a graph consists of vertices and directed edges, with no edge from a vertex to itself and no edges pointing both ways between a pair. The claim guarantees a vertex with at least as many vertices at directed distance exactly two as at distance one. Distance means the length of a shortest path following the arrows.

What does that help mathematicians do?

The claimed result would rule out any such graph in which every vertex's second neighborhood is strictly smaller than its first. More concretely, if every vertex has at least d outgoing neighbors, some vertex must have at least d additional vertices at distance exactly two. These are distinct from its immediate neighbors. This supplies a guaranteed local configuration for counting arguments without assumptions about how densely connected the graph is.

Are there practical applications?

The immediate value is foundational: the claim would give researchers a general constraint on how one-step and two-step neighborhoods compare in oriented graphs. Directed networks provide an intuitive model, but the guarantee concerns a graph structure, not a demonstrated practical procedure. It does not by itself establish an efficient way to find the promised vertex.

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.

Manuscript

A proof of Seymour’s second-neighborhood conjecture

September 23, 2026 15 pages Main result formalized in Lean

We prove that every nonempty finite oriented graph has a vertex with at least as many vertices at directed distance two as at directed distance one. This resolves Seymour's second neighborhood conjecture positively.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026,
  author = {{OpenAI}},
  title = {{A proof of Seymour's second-neighborhood conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026/paper.pdf}{OAI:A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/173.md.

Seymour’s second-neighborhood conjecture

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

Scope

Seymour's second-neighborhood conjecture asserts that every nonempty finite oriented graph has a vertex with at least as many second out-neighbors as first out-neighbors. The formalized result proves this assertion, where the second neighborhood consists of vertices at directed distance exactly two. The initial vertex and first neighbors are excluded from that count, and sinks are included.

Comparator links

Result Comparator statement
Seymour's second-neighborhood conjecture SeymourSecondNeighborhood.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.