Erdős Problem #61
The Erdős–Hajnal Conjecture states that there is a constant c(H) > 0 for each H such that we can take f(n) = n^c(H) in the above formulation.
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.
Cite
@misc{cairn-erdos-61,
title = {Erdős Problem #61},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/erdos-61}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
} Also: CITATION.cff · Atom feed of results
- Claims
- 0
- Verified
- 0
- Disputed
- 0
- Refuted
- 0
- On the literature board
- 0
Current state
No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.
The problem
The question
The Erdős–Hajnal Conjecture states that there is a constant for each such that we can take in the above formulation.
Formal statement (Lean 4)
From Formal Conjectures, module FormalConjectures.ErdosProblems.«61». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.
theorem erdos_61 :
answer(sorry) ↔ ∀ {α : Type*} [Fintype α] [DecidableEq α] (H : SimpleGraph α),
∃ c > (0 : ℝ), IsErdosHajnalLowerBound H (fun n : ℕ => (n : ℝ) ^ c)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/61. Report it as a literature claim.
- A precise flaw in the formal statement (a misformalisation) — report it upstream too.
References
Source and licence
Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.