Erdős Problem #1154
Does there exist, for every α ∈ [0, 1], a ring in ℝ with Hausdorff dimension α?
Current state (tracked automatically)
- erdosproblems.com lists Erdős problem #1154 as not disprovable (status updated 24 January 2026).
All tracked facts and sources
| Fact | Value | Source |
|---|---|---|
| Status of Erdős problem #1154 on erdosproblems.com | not disprovable 2026-01-24 | erdosproblems.com |
| Formal Conjectures: research statements in the file | 2 open, 3 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-1154,
title = {Erdős Problem #1154},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/erdos-1154}},
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-1154)
- 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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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.
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Current state
No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.
The problem
The question
erdos_1154.parts.i. Does there exist, for every , a ring in with Hausdorff dimension ?
erdos_1154.parts.ii. Does there exist, for every , a field in with Hausdorff dimension ?
Formal statement (Lean 4)
From Formal Conjectures, module FormalConjectures.ErdosProblems.«1154» (2 statements).
theorem erdos_1154.parts.i :
∀ a ∈ Icc (0 : ℝ≥0∞) 1, ∃ R : Subring ℝ, dimH (R : Set ℝ) = atheorem erdos_1154.parts.ii :
∀ a ∈ Icc (0 : ℝ≥0∞) 1, ∃ R : Subfield ℝ, dimH (R : Set ℝ) = aWhat 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/1154. Report it as a literature claim.
- A precise flaw in the formal statement (a misformalisation) — report it upstream too.
References
- erdosproblems.com/1154
- [ErVo66] Erdős, Paul and Volkmann, Bodo, Additive Gruppen mit vorgegebener Hausdorffscher Dimension. J. Reine Angew. Math. (1966), 203--208.
- [EdMi01] Edgar, G. A. and Miller, Chris, Hausdorff dimension, analytic sets and transcendence. Real Anal. Exchange (2001/02), 335--339.
- [EdMi03] Edgar, G. A. and Miller, Chris, Borel subrings of the reals. Proc. Amer. Math. Soc. (2003), 1121--1129.
- [Ma16b] Mauldin, R. Daniel, Subfields of ℝ with arbitrary Hausdorff dimension. Math. Proc. Cambridge Philos. Soc. (2016), 157--165.
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.