1159.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 1159

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.

Imports (1)

FormalConjecturesUtil

theorem Erdos1159.erdos_1159 Determine whether there exists a constant $C>1$ such that the following holds sorry line 43 source
Source documentation
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 $1\leq \lvert S\cap \ell\rvert \leq C$ for all lines $\ell$?

L4DEL explanation
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 $1\leq \lvert S\cap \ell\rvert \leq C$ for all lines $\ell$?
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, And, LT.lt, OfNat.ofNat, Membership, Fintype, Configuration.ProjectivePlane, Set, LE.le, Set.ncard, Inter.inter, Set.ofPred, Membership.mem
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
8 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
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 := by
…
```