FormalConjectures.ErdosProblems.«1207»

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module

public import FormalConjecturesUtil

Erdős Problem 1207

References:

erdosproblems.com/1207

[BMP05] Brass, Peter and Moser, William and Pach, János, Research problems in discrete geometry. (2005), xii+499.

[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.

[PaTa02] Pach, János and Tardos, Gábor, Isosceles triangles determined by a planar point set. Graphs Combin. (2002), 769--779.

@[expose] public sectionopen Filteropen scoped EuclideanGeometrynamespace Erdos1207

P d n is the largest number $m$ such that every set of $n$ points in $\mathbb{R}^d$ has an isosceles-free subset of size at least $m$.

noncomputable def P (d n : ℕ) : ℕ :=
  sInf {m : ℕ | ∃ S : Finset (ℝ^d), S.card = n ∧
    m = sSup {k : ℕ | ∃ A ⊆ S, (A : Set (ℝ^d)).IsIsoscelesFree ∧ A.card = k}}

Let $P_d(n)$ be such that in any set of $n$ points in $\mathbb{R}^d$ there exist at least $P_d(n)$ many points which do not contain an isosceles triangle. Estimate $P_d(n)$ - in particular, is it true that $$P_2(n)<n^{1-c}$$ for some constant $c>0$?

@[category research open, AMS 52]
theorem erdos_1207 : answer(sorry) ↔
    ∃ c > (0 : ℝ), ∀ᶠ n : ℕ in atTop, (P 2 n : ℝ) < (n : ℝ) ^ (1 - c) := by⊢ True ↔ ∃ c > 0, ∀ᶠ (n : ℕ) in atTop, ↑(P 2 n) < ↑n ^ (1 - c)
  sorryAll goals completed! 🐙

end Erdos1207