Erdős (Erdos) Problem #838: status in 2026

Erdős Problem #838

Let f(n) be maximal such that any n points in ℝ^2, with no three on a line, determine at least f(n) different convex subsets. Estimate f(n) - in particular, does there exist a constant c such that lim log f(n)/(log n)^2=c? A question of Erdős and Hammer [Er78c, pp. 52-53]. This is the first part.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #838 as open (status updated 31 August 2025).

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #838 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: research statements in the file	2 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.

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Cite

@misc{cairn-erdos-838,
  title        = {Erdős Problem #838},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-838}},
  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_838.parts.i. Let $f (n)$ be maximal such that any $n$ points in $R^{2}$ , with no three on a line, determine at least $f (n)$ different convex subsets. Estimate $f (n)$ - in particular, does there exist a constant $c$ such that $lim \frac{lo g f ( n )}{( lo g n ) ^{2}} = c ?$

A question of Erdős and Hammer [Er78c, pp. 52-53]. This is the first part. We read "estimate" as asking for an asymptotic formula for $lo g f (n)$ .

erdos_838.parts.ii. Let $f (n)$ be maximal such that any $n$ points in $R^{2}$ , with no three on a line, determine at least $f (n)$ different convex subsets. Estimate $f (n)$ - in particular, does there exist a constant $c$ such that $lim \frac{lo g f ( n )}{( lo g n ) ^{2}} = c ?$

A question of Erdős and Hammer [Er78c, p. 53]. This is the second part.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«838» (2 statements).

theorem erdos_838.parts.i :
    (fun n : ℕ ↦ Real.log (f n)) ~[atTop] erdos_838.parts.i.answer

theorem erdos_838.parts.ii :
    ∃ c : ℝ, Tendsto (fun n : ℕ ↦ Real.log (f n) / Real.log n ^ 2) atTop (nhds c)

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/838. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/838
[Er78c] Erdős, P., Some more problems on elementary geometry. Austral. Math. Soc. Gaz. 5 (1978), 52-54. PDF

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.