Erdős (Erdos) Problem #1158: status in 2026

Erdős Problem #1158

Let K_t(r) be the complete t-partite t-uniform hypergraph with r vertices in each class. Is it true that ex_t(n,K_t(r)) ≥ n^t-r^1-t-o(1) for all t,r≥ 2? The restriction r≥ 2 excludes forbidding a single edge.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2026-01-23).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #1158 as open (status updated 23 January 2026).
Formal Conjectures marks the Lean statement erdos_1158 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #1158 on erdosproblems.com	open 2026-01-23	erdosproblems.com
Formal Conjectures: status of `erdos_1158`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 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. 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-1158,
  title        = {Erdős Problem #1158},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1158}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

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Claims

Verified

Disputed

Refuted

On the literature board

Nobody has worked on this problem here yet

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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 $K_{t} (r)$ be the complete $t$ -partite $t$ -uniform hypergraph with $r$ vertices in each class. Is it true that $ex_{t} (n, K_{t} (r)) \geq n^{t - r^{1 - t} - o (1)}$ for all $t, r \geq 2$ ? The restriction $r \geq 2$ excludes forbidding a single edge.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1158».

theorem erdos_1158 :
    ∀ t r : ℕ, 2 ≤ t → 2 ≤ r →
    ∀ ε : ℝ, 0 < ε → ∀ᶠ n : ℕ in atTop,
      (n : ℝ) ^ ((t : ℝ) - (r : ℝ) ^ (1 - (t : ℝ)) - ε) ≤
        (Hypergraph.partiteExtremalNumber n t r : ℝ)

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/1158. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/1158
[Er64f] Erdős, P., On extremal problems of graphs and generalized graphs. Israel J. Math. (1964), 183--190.

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.