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import FormalConjecturesUtilErdős Problem 757
References:
[GyLe95] Gyárfás, András and Lehel, Jenő, Linear sets with five distinct differences among any four elements. J. Combin. Theory Ser. B (1995), 108-118.
open scoped Pointwiseopen Filternamespace Erdos757
We say that c is admissible if, for any finite set A such that every four-element
subset B determines at least five positive differences, there exists a Sidon subset S
of size at least c * A.ncard. The difference condition is equivalent to
11 ≤ (B - B).ncard.
def IsAdmissible (c : ℝ) : Prop := ∀ {A : Set ℝ}, A.Finite → (∀ B ⊆ A,
B.ncard = 4 → 11 ≤ (B - B).ncard) → ∃ S ⊆ A, IsSidon S ∧ c * A.ncard ≤ (S.ncard : ℝ)What is the supremum of the set of admissible numbers?
@[category research open, AMS 5]
theorem erdos_757 {A : Set ℝ} :
answer(sorry) = sSup {c | IsAdmissible c} := A:Set ℝ⊢ sorry = sSup {c | IsAdmissible c}
All goals completed! 🐙
The supremum is strictly larger than 1 / 2, which is proved in [GyLe95].
@[category research solved, AMS 5]
theorem erdos_757.variants.lowerBound {A : Set ℝ} : 1 / (2 : ℝ) < sSup {c | IsAdmissible c} := A:Set ℝ⊢ 1 / 2 < sSup {c | IsAdmissible c}
All goals completed! 🐙
In [GyLe95], the authors also prove that the supremum is smaller than 3 / 5.
@[category research solved, AMS 5]
theorem erdos_757.variants.upperBound {A : Set ℝ} : sSup {c | IsAdmissible c} < 3 / (5 : ℝ) := A:Set ℝ⊢ sSup {c | IsAdmissible c} < 3 / 5
All goals completed! 🐙end Erdos757