The Gyárfás–Sumner conjecture

The Gyárfás–Sumner conjecture (Gyárfás 1975, Sumner 1981). For every finite tree T the class of T-free graphs (graphs with no induced subgraph isomorphic to T) is χ-bounded: there is a function f such that every T-free graph G satisfies χ(G) ≤ f(ω(G)).

Current state (tracked automatically)

Formal Conjectures marks the Lean statement gyarfas_sumner_conjecture as open.

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: status of `gyarfas_sumner_conjecture`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 2 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Formal Conjectures (checked 2026-11-12). How it works.

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-gyarfas-sumner-conjecture,
  title        = {The Gyárfás–Sumner conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/gyarfas-sumner-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

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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

The Gyárfás–Sumner conjecture (Gyárfás 1975, Sumner 1981).

For every finite tree $T$ the class of $T$ -free graphs (graphs with no induced subgraph isomorphic to $T$ ) is $χ$ -bounded: there is a function $f$ such that every $T$ -free graph $G$ satisfies $χ (G) \leq f (ω (G))$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.GyarfasSumnerConjecture.

theorem gyarfas_sumner_conjecture :
    ∀ {W : Type} [Fintype W] [DecidableEq W] (T : SimpleGraph W) [DecidableRel T.Adj],
      T.IsTree → ∃ f : ℕ → ℕ, IsChiBoundedBy T f

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.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Wikipedia
[Gy75] Gyárfás, A. (1975). "On Ramsey covering-numbers." Infinite and finite sets, Colloq. Math. Soc. János Bolyai 10, pp. 801--816.
[Su81] Sumner, D. P. (1981). "Subtrees of a graph and the chromatic number." *The theory and applications of graphs (Kalamazoo, 1980)*, Wiley, pp. 557--576.
[Gy87] Gyárfás, A. (1987). "Problems from the world surrounding perfect graphs." Zastos. Mat. 19, pp. 413--441.
[KP94] Kierstead, H. A. and Penrice, S. G. (1994). "Radius two trees specify χ-bounded classes." J. Graph Theory 18, pp. 119--129.
[SS20] Scott, A. and Seymour, P. (2020). "A survey of χ-boundedness." J. Graph Theory 95, pp. 473--504. arXiv:1812.07500

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.