Erdős Problem #918
Erdős Problem #918
Current state (tracked automatically)
- erdosproblems.com lists Erdős problem #918 as open (status updated 31 August 2025).
All tracked facts and sources
| Fact | Value | Source |
|---|---|---|
| Status of Erdős problem #918 on erdosproblems.com | open 2025-08-31 | erdosproblems.com |
| Formal Conjectures: research statements in the file | 4 open, 1 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.
Cite
@misc{cairn-erdos-918,
title = {Erdős Problem #918},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/erdos-918}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
} Also: CITATION.cff · Atom feed of results
Status badge for a README (shields.io):
[](https://cairn-commons.com/problems/erdos-918)
- Claims
- 0
- Verified
- 0
- Disputed
- 0
- Refuted
- 0
- On the literature board
- 0
Nobody has worked on this problem here yet
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- Get a task — one click
- Your chatbot works on it — 5–10 minutes, in its own tab
- Paste its answer here — check it and publish it under your name
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.
ChatGPT opens in a new tab and starts on the task right away; Claude opens with it filled in.
Current state
No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.
The problem
The question
erdos_918.parts.i. (no description upstream)
erdos_918.parts.ii. (no description upstream)
Formal statement (Lean 4)
From Formal Conjectures, module FormalConjectures.ErdosProblems.«918» (2 statements).
theorem erdos_918.parts.i :
∃ (V : Type u) (G : SimpleGraph V), #V = ℵ_ 2 ∧ G.chromaticCardinal = ℵ_ 2 ∧
∀ (W : Set V) (_ : #W = ℵ₁), (G.induce W).chromaticCardinal ≤ ℵ₀theorem erdos_918.parts.ii :
∃ (V : Type u) (G : SimpleGraph V), #V = ℵ_ (ω + 1) ∧ G.chromaticCardinal = ℵ₁ ∧
∀ (W : Set V) (_ : #W = ℵ_ ω), (G.induce W).chromaticCardinal ≤ ℵ₀What 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/918. Report it as a literature claim.
- A precise flaw in the formal statement (a misformalisation) — report it upstream too.
Variants
erdos_918.variants.all_subgraphs.parts.ierdos_918.variants.all_subgraphs.parts.ii— Is there a graph with aleph_ω+1 vertices and chromatic number aleph_1 such that every subgraph on aleph_ω vertices has chromatic number ≤aleph_0?
References
- erdosproblems.com/918
- [ErHa68b] Erdős, P. and Hajnal, A., On chromatic number of infinite graphs. (1968), 83--98.
- [Er69b] Erdős, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968) (1969), 27-35.
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.