Erdős (Erdos) Problem #1017: status in 2026

Erdős Problem #1017

Let f(n,k) be such that every graph on n vertices and k edges can be partitioned into at most f(n,k) edge-disjoint complete graphs. Estimate f(n,k) for k>n^2/4.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-09-12).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #1017 as open (status updated 12 September 2025).

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #1017 on erdosproblems.com	open 2025-09-12	erdosproblems.com
Formal Conjectures: research statements in the file	2 open, 0 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), 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.

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Cite

@misc{cairn-erdos-1017,
  title        = {Erdős Problem #1017},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1017}},
  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

erdos_1017.lower_bound. Let $f (n, k)$ be such that every graph on $n$ vertices and $k$ edges can be partitioned into at most $f (n, k)$ edge-disjoint complete graphs. Estimate $f (n, k)$ for $k > n^{2} /4$ .

erdos_1017.upper_bound. Estimate an upper bound for $f (n, k)$ , the worst-case number of complete subgraphs needed to partition the edges of an $n$ -vertex, $k$ -edge graph, for $n^{2} /4 < k \leq (2 n)$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1017» (2 statements).

theorem erdos_1017.lower_bound :
    ∀ n k : ℕ, k ≤ n.choose 2 → n * n < 4 * k →
      erdos_1017.lower_bound.answer n k ≤ f n k

theorem erdos_1017.upper_bound :
    ∀ n k : ℕ, k ≤ n.choose 2 → n * n < 4 * k →
      f n k ≤ erdos_1017.upper_bound.answer n k

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
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/1017. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/1017
[EGP66] Erdős, Paul and Goodman, A. W. and Pósa, Lajos, The representation of a graph by set intersections. Canadian J. Math. (1966), 106–112.
[Er71] Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97–109.
[GyKe17] Győri, Ervin and Keszegh, Balázs, On the number of edge-disjoint triangles in $K_{4}$ -free graphs. Combinatorica (2017), 1113–1124.
[Lo68] Lovász, L., On covering of graphs. Theory of Graphs (Proc. Colloq., Tihany, 1966) (1968), 231–236.

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.