Erdős (Erdos) Problem #743: status in 2026

Erdős Problem #743

Let T_2,…,T_n be a collection of trees such that T_k has k vertices. Can we always write K_n as the edge disjoint union of the T_k? A conjecture of Gyárfás, known as the tree packing conjecture. We state it as a packing.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31). It is marked falsifiable there: if it is false, a finite computation can show it.

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-erdos-743,
  title        = {Erdős Problem #743},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-743}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

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

Let $T_{2}, \dots, T_{n}$ be a collection of trees such that $T_{k}$ has $k$ vertices. Can we always write $K_{n}$ as the edge disjoint union of the $T_{k}$ ?

A conjecture of Gyárfás, known as the tree packing conjecture. We state it as a packing. For every $n$ and all trees $T_{k}$ on the vertex set Fin k, $2 \leq k \leq n$ , there are injective maps $f_{k} : {0, \dots, k - 1} \to {0, \dots, n - 1}$ such that the graphs $f_{k} (T_{k})$ are pairwise edge-disjoint (see IsPacking). Every tree with $k$ vertices is isomorphic to a tree on Fin k, so this covers every collection of trees.

A tree with $k$ vertices has $k - 1$ edges, and $\sum_{k = 2}^{n} (k - 1) = (2 n)$ is the number of edges of $K_{n}$ . Hence the copies $f_{k} (T_{k})$ of a packing cover every edge of $K_{n}$ . So a packing is the same as a decomposition of $K_{n}$ into copies of $T_{2}, \dots, T_{n}$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«743».

theorem erdos_743 :
    ∀ n : ℕ, ∀ T : (k : ℕ) → SimpleGraph (Fin k),
      (∀ k ∈ Set.Icc 2 n, (T k).IsTree) → IsPacking n (Set.Icc 2 n) T

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 — check erdosproblems.com/743. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/743
[ABCHPT21] Allen, Peter and Böttcher, Julia and Clemens, Dennis and Hladký, Jan and Piguet, Diana and Taraz, Anusch, *The tree packing conjecture for trees of almost linear maximum degree*. arXiv:2106.11720 (2021).
[Bo83] Bollobás, Béla, Some remarks on packing trees. Discrete Math. (1983), 203-204.
[Er81] Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.
[Fi83] Fishburn, P. C., Balanced integer arrays: a matrix packing theorem. J. Combin. Theory Ser. A (1983), 98-101.
[Fi83b] Fishburn, P. C., Packing graphs with odd and even trees. J. Graph Theory (1983), 369-383.
[GyLe78] Gyárfás, A. and Lehel, J., Packing trees of different order into $K_{n}$ . Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Colloq. Math. Soc. János Bolyai 18 (1978), 463-469.
[JKKO19] Joos, Felix and Kim, Jaehoon and Kühn, Daniela and Osthus, Deryk, *Optimal packings of bounded degree trees*. J. Eur. Math. Soc. (JEMS) (2019), 3573-3647.
[JaMo24] Janzer, Barnabás and Montgomery, Richard, *Packing the largest trees in the tree packing conjecture*. arXiv:2403.10515 (2024).

Source and licence

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