Erdős Problem #1159 — open problem in combinatorics

Erdős Problem #1159

Determine whether there exists a constant C>1 such that the following holds. Let P be a finite projective plane. Must there exist a set of points S such that 1≤ lvert S∩ ℓrvert ≤ C for all lines ℓ?

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-1159,
  title        = {Erdős Problem #1159},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1159}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Determine whether there exists a constant $C > 1$ such that the following holds.

Let $P$ be a finite projective plane. Must there exist a set of points $S$ such that $1 \leq ∣ S \cap ℓ ∣ \leq C$ for all lines $ℓ$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1159». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_1159 : answer(sorry) ↔
    ∃ C : ℕ, 1 < C ∧
      ∀ (P L : Type) (_ : Membership P L) (_ : Fintype P) (_ : Fintype L),
        ∀ _ : ProjectivePlane P L, ∃ S : Set P, ∀ l : L,
          1 ≤ (S ∩ {p : P | p ∈ l}).ncard ∧ (S ∩ {p : P | p ∈ l}).ncard ≤ 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/1159. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/1159
[ESS83] Erdős, P. and Silverman, R. and Stein, A., *Intersection properties of families containing sets of nearly the same size*. Ars Combin. (1983), 247--259.
[Er81] Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.

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.