Sumner's universal tournament conjecture (1971)

Sumner's universal tournament conjecture (1971). Every tournament on 2n - 2 vertices contains every oriented tree on n ≥ 2 vertices as a subgraph.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-sumner-universal-tournament-conjecture,
  title        = {Sumner's universal tournament conjecture (1971)},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/sumner-universal-tournament-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

Sumner's universal tournament conjecture (1971).

Every tournament on $2 n - 2$ vertices contains every oriented tree on $n \geq 2$ vertices as a subgraph.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.SumnerUniversalTournamentConjecture.

theorem sumner_conjecture :
    ∀ (n : ℕ), 2 ≤ n →
      ∀ {W : Type} [Fintype W] [DecidableEq W] (D : Digraph W) [DecidableRel D.Adj],
        Fintype.card W = n → IsOrientedTree D →
        ∀ {V : Type} [Fintype V] [DecidableEq V] (G : Digraph V) [DecidableRel G.Adj],
          Fintype.card V = 2 * n - 2 → G.IsTournament → Embeds D G

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
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References

Wikipedia
[HT91] Häggkvist, R. and Thomason, A. (1991). "Trees in tournaments." Combinatorica 11, pp. 123--130.
[ES04] El Sahili, A. (2004). "Trees in tournaments." J. Combin. Theory Ser. B 92, pp. 183--187.
[KMO11] Kühn, D., Mycroft, R. and Osthus, D. (2011). "A proof of Sumner's universal tournament conjecture for large tournaments." Proc. London Math. Soc. 102, pp. 731--766. arXiv:1010.4430

Source and licence

Imported from Formal Conjectures (Wikipedia), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.