FormalConjectures.ErdosProblems.«1159»

/-
Copyright 2026 The Formal Conjectures Authors.

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

    https://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/
module

public import FormalConjecturesUtil

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.

@[expose] public sectionopen Configurationnamespace Erdos1159

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 $\ell$?

@[category research open, AMS 5 51]
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⊢ True ↔
  ∃ C,
    1 < C ∧
      ∀ (P L : Type) (x : Membership P L) (x_1 : Fintype P) (x_2 : Fintype L) (x_3 : ProjectivePlane P L),
        ∃ S, ∀ (l : L), 1 ≤ (S ∩ {p | p ∈ l}).ncard ∧ (S ∩ {p | p ∈ l}).ncard ≤ C
  sorryAll goals completed! 🐙

end Erdos1159