1 · The essentials
Imagine a finite network of points joined by one-way arrows. No arrow goes from a point back to itself, and no pair of points has arrows in both directions. From a chosen point, count the points you can reach by following one arrow. Then count the points whose shortest route from that starting point requires exactly two arrows. The result says that, as long as the network has at least one point, some starting point has a second count at least as large as its first count. This is the statement of Seymour’s second-neighborhood conjecture, and the supplied source reports a Lean-formalized proof.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, picture a finite collection of places connected by one-way routes, with no direct return route between any connected pair. The result guarantees a place from which at least as many destinations require exactly two route segments as require one; the image is imperfect because an actual route map need not obey the stated restrictions.
Why it matters
The result supplies a universal guarantee about these networks without imposing any further restrictions on their arrows. The source presents it as a positive resolution of Seymour’s second-neighborhood conjecture, with a Lean formalization of the main result.
2 · Going further
An oriented graph here is a collection of vertices and directed edges with no loops and no oppositely directed edge pairs. For a vertex , write for the vertices at directed distance one from , and for those at directed distance exactly two. The precise assertion is that every nonempty finite oriented graph has a vertex satisfying . In particular, a vertex already reachable in one step is not counted at distance exactly two merely because a two-step route also reaches it.
The repository summary and the main manuscript’s abstract both state this existence result. The abstract describes it as a positive resolution of Seymour’s second-neighborhood conjecture. The hypotheses allow arbitrary oriented graphs apart from finiteness, nonemptiness, and the exclusions of loops and oppositely directed edge pairs.
The main manuscript is identified as Lean-formalized. However, the supplied abstract gives no intermediate lemmas, construction, or other description of the argument, so its proof strategy cannot be explained from this source alone. Only one manuscript is supplied; there are no companion manuscripts here whose relationship to the main result can be described.
Useful notions
- Finite sets and their sizes
- Vertices and directed edges
- Following a directed path
- Shortest-path distance
What was known before / what is new
Before
The source identifies the statement as Seymour’s second-neighborhood conjecture. It provides no account of earlier partial results, methods, or historical development.
New
The main manuscript states the result for every nonempty finite oriented graph, rather than for a restricted subclass. The source reports that this resolves the conjecture positively and identifies the main result as Lean-formalized.
Limits and assumptions
The guarantee requires a finite graph with at least one vertex, no loops, and no oppositely directed edge pairs. It asserts the existence of at least one suitable vertex, not that every vertex satisfies the inequality. The second count concerns shortest directed distance exactly two. The main result has a Lean formalization, but the supplied text does not describe the formalization’s internal details or the proof strategy. It lists no remaining open questions.
Mini glossary
- Oriented graph —
- A graph with directed edges, no edge from a vertex to itself, and no pair of edges pointing in opposite directions between the same two vertices.
- Directed distance —
- The smallest number of edges in a route from one vertex to another that follows the direction of every edge.
- First out-neighborhood —
- The set of vertices reachable from a chosen vertex by one directed edge.
- Second out-neighborhood —
- The set of vertices whose shortest directed route from a chosen vertex has exactly two edges.
- Lean formalization —
- An encoding of a mathematical statement and its proof in Lean for formal checking.
Keywords : Oriented graphs · Second neighborhoods · Seymour’s conjecture
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
A proof of Seymour’s second-neighborhood conjecture
23/09/2026We 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.