FormalConjectures.ErdosProblems.«2»

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import FormalConjecturesUtil

Erdős Problem 2

Reference: erdosproblems.com/2

Erdős asked whether the smallest modulus in a distinct covering system can be arbitrarily large. Hough proved that the answer is no, and Balister, Bollobás, Morris, Sahasrabudhe, and Tiba later gave a simpler proof with an improved explicit upper bound.

namespace Erdos2

Can the smallest modulus of a covering system be arbitrarily large?

This problem has a negative answer: there is a universal bound on the least modulus of any distinct covering system.

@[category research solved, AMS 11]
theorem erdos_2 :
    answer(False) ↔
      ∀ B : ℕ, ∃ c : StrictCoveringSystem ℤ, ∀ i, ∃ m : ℕ,
        c.moduli i = Ideal.span {(m : ℤ)} ∧ B < m := by⊢ False ↔ ∀ (B : ℕ), ∃ c, ∀ (i : c.ι), ∃ m, c.moduli i = Ideal.span {↑m} ∧ B < m
  sorryAll goals completed! 🐙

end Erdos2