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import FormalConjecturesUtil
open Filteropen EuclideanGeometrynamespace Erdos107The set of $N$ such that any $N$ points in the plane, no three on a line, contain a convex $n$-gon.
def cardSet (n : ℕ) := { N | ∀ (pts : Finset ℝ²), pts.card = N → NonTrilinear (pts : Set ℝ²) →
HasConvexNGon n pts }
The function $f(n)$ specified in erdos_107.
noncomputable def f (n : ℕ) : ℕ :=
sInf (cardSet n)Let $f(n)$ be minimal such that any $f(n)$ points in $ℝ^2$, no three on a line, contain $n$ points which form the vertices of a convex $n$-gon. Prove that $f(n) = 2^{n-2} + 1$.
@[category research open, AMS 52]
theorem erdos_107 : answer(sorry) ↔ ∀ n ≥ 3, f n = 2^(n - 2) + 1 := ⊢ True ↔ ∀ n ≥ 3, f n = 2 ^ (n - 2) + 1
All goals completed! 🐙For every $n ≥ 3$, there exists $N$ such that any $N$ points, no three on a line, contain a convex $n$-gon.
@[category research solved, AMS 52]
theorem nonempty_cardSet : ∀ n ≥ 3, (cardSet n).Nonempty := ⊢ ∀ n ≥ 3, (cardSet n).Nonempty
All goals completed! 🐙Depending on details of definitions, the statement is false or trivial for $n < 3$.
this:∀ (P : Set ℝ²), HasConvexNGon 0 P⊢ f 0 = 0
All goals completed! 🐙Three distinct non-collinear points form a convex-independent set.
a:ℝ²b:ℝ²hab:a ≠ bx:ℝ²hac:a ≠ xhbc:b ≠ xhcoll:¬Collinear ℝ {a, b, x}hmem:x ∈ (convexHull ℝ) ({a, b, x} \ {x})hx_aff:x ∈ affineSpan ℝ ({a, b, x} \ {x})⊢ Collinear ℝ {a, b}
All goals completed! 🐙hmem3:3 ∈ cardSet 3N:ℕhN:N ∈ cardSet 3hlt:N < 3f:ℕ → ℝ² := fun i ↦ EuclideanSpace.single 0 ↑ipts:Finset ℝ² := Finset.image f (Finset.range N)hinj:Function.Injective fhpts_card:pts.card = Nhnontri:NonTrilinear ↑ptsS:Finset ℝ²hScard:S.card = 3hSsub:↑S ⊆ ↑ptsright✝:ConvexIndep ↑Sthis:S.card ≤ pts.card⊢ False
All goals completed! 🐙namespace variantsErdős and Szekeres proved the bounds $$ 2^{n-2} + 1 ≤ f(n) ≤ \binom{2n-4}{n-2} + 1 $$ ([ErSz60] and [ErSz35] respectively).
[ErSz60] Erdős, P. and Szekeres, G., On some extremum problems in elementary geometry. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. (1960/61), 53-62.
[ErSz35] Erdős, P. and Szekeres, G., A combinatorial problem in geometry. Compos. Math. (1935), 463-470.
@[category research solved, AMS 52]
theorem ersz_bounds :
∀ n ≥ 3, 2^(n - 2) + 1 ≤ f n ∧ f n ≤ Nat.choose (2 * n - 4) (n - 2) + 1 := ⊢ ∀ n ≥ 3, 2 ^ (n - 2) + 1 ≤ f n ∧ f n ≤ (2 * n - 4).choose (n - 2) + 1
All goals completed! 🐙Suk [Su17] proved $$ f(n) ≤ 2^{(1+o(1))n}. $$
[Su17] Suk, Andrew, On the Erdős-Szekeres convex polygon problem. J. Amer. Math. Soc. (2017), 1047-1053.
@[category research solved, AMS 52]
theorem su_bound :
∃ r : ℕ → ℝ, r =o[atTop] (fun n => (n : ℝ)) ∧
∀ n ≥ 3, (f n : ℝ) ≤ 2^(n + r n) := ⊢ ∃ r, (r =o[atTop] fun n ↦ ↑n) ∧ ∀ n ≥ 3, ↑(f n) ≤ 2 ^ (↑n + r n)
All goals completed! 🐙The current best bound is due to Holmsen, Mojarrad, Pach, and Tardos [HMPT20], who prove $$ f(n) ≤ 2^{n+O(\sqrt{n\log n})}. $$
[HMPT20] Holmsen, Andreas F. and Mojarrad, Hossein Nassajian and Pach, János and Tardos, Gábor, Two extensions of the Erdős-Szekeres problem. J. Eur. Math. Soc. (JEMS) (2020), 3981-3995.
@[category research solved, AMS 52]
theorem hmpt_bound :
∃ r : ℕ → ℝ, r =O[atTop] (fun n => Real.sqrt (n * Real.log n)) ∧
∀ n ≥ 3, (f n : ℝ) ≤ 2^(n + r n) := ⊢ ∃ r, (r =O[atTop] fun n ↦ √(↑n * Real.log ↑n)) ∧ ∀ n ≥ 3, ↑(f n) ≤ 2 ^ (↑n + r n)
All goals completed! 🐙end Erdos107.variants