107.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

10 declarations, 5 contain sorry, 0 declared as axioms

Erdős Problem 107

References:

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Wikipedia/HappyEndingProblem.lean

def Erdos107.cardSet The set of $N$ such that any $N$ points in the plane, no three on a line, contain a convex $n$-gon line 37 source
Source documentation
The set of $N$ such that any $N$ points in the plane, no three on a line, contain a convex $n$-gon.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set
Proof uses: Set.ofPred, Finset, EuclideanSpace, Real, Fin, OfNat.ofNat, Eq, Finset.card, EuclideanGeometry.NonTrilinear, ENNReal, EuclideanGeometry.HasConvexNGon
24 instances, coercions and auxiliary constants left out.
```
def cardSet (n : ℕ) := { N | ∀ (pts : Finset ℝ²), pts.card = N → NonTrilinear (pts : Set ℝ²) →
…
```
def Erdos107.f The function $f(n)$ specified in erdos_107 line 41 source
Source documentation
The function $f(n)$ specified in erdos_107.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: InfSet.sInf, Erdos107.cardSet
1 instances, coercions and auxiliary constants left out.
```
noncomputable def f (n : ℕ) : ℕ :=
…
```
theorem Erdos107.erdos_107 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 sorry line 50 source
Source documentation
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 $f(n) = 2^{n-2} + 1$.

depends on sorry what this means
Statement uses: Iff, True, Nat, GE.ge, OfNat.ofNat, Eq, Erdos107.f, HAdd.hAdd, HPow.hPow, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_107 : answer(sorry) ↔ ∀ n ≥ 3, f n = 2^(n - 2) + 1 := by
…
```
Mentions: Erdos107.f
theorem Erdos107.nonempty_cardSet For every $n ≥ 3$, there exists $N$ such that any $N$ points, no three on a line, contain a convex $n$-gon sorry line 56 source
Source documentation
For every $n ≥ 3$, there exists $N$ such that any $N$ points, no three on a line, contain a convex $n$-gon.

depends on sorry what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, Set.Nonempty, Erdos107.cardSet
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem nonempty_cardSet : ∀ n ≥ 3, (cardSet n).Nonempty := by
…
```
Mentions: Erdos107.cardSet
theorem Erdos107.f_zero_eq Depending on details of definitions, the statement is false or trivial for $n < 3$ line 62 source
Source documentation
Depending on details of definitions, the statement is false or trivial for $n < 3$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Erdos107.f, OfNat.ofNat
Proof uses: Set, EuclideanSpace, Real, Fin, EuclideanGeometry.HasConvexNGon, Exists.intro, Finset, And, Finset.card, LE.le, EuclideanGeometry.ConvexIndep, EmptyCollection.emptyCollection, of_eq_true, Eq.trans, True, congr, congrArg, eq_self, congrFun', Finset.coe_empty and 25 more
63 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
theorem f_zero_eq : f 0 = 0 := by
…
```
Mentions: Erdos107.f

lemma Erdos107.convexIndep_triple_of_not_collinear Three distinct non-collinear points form a convex-independent set line 69 source

Source documentation

Three distinct non-collinear points form a convex-independent set.

private lemma convexIndep_triple_of_not_collinear {a b c : ℝ²}
    (hab : a ≠ b) (hac : a ≠ c) (hbc : b ≠ c)
    (hcoll : ¬ Collinear ℝ ({a, b, c} : Set ℝ²)) :
    ConvexIndep ({a, b, c} : Set ℝ²) := by
…

Mentions: c

theorem Erdos107.f_three_eq line 93 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Erdos107.f, OfNat.ofNat
Proof uses: Membership.mem, Set, Erdos107.cardSet, Finset, EuclideanSpace, Real, Fin, Finset.card, EuclideanGeometry.NonTrilinear, ENNReal, Exists, And, Ne, Insert.insert, Singleton.singleton, EuclideanGeometry.HasConvexNGon, Iff.mp, Finset.card_eq_three, Exists.intro, LE.le and 49 more
126 instances, coercions and auxiliary constants left out.

Tactic trace: 31 steps
Open to load the checked before-and-after goal states.
theorem Erdos107.variants.ersz_bounds Erdős and Szekeres proved the bounds $$ 2^{n-2} + 1 ≤ f(n) ≤ \binom{2n-4}{n-2} + 1 $$ ([ErSz60] and [ErSz35] respectively) sorry line 139 source
Source documentation
Erdős 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.

depends on sorry what this means
Statement uses: Nat, GE.ge, OfNat.ofNat, And, LE.le, HAdd.hAdd, HPow.hPow, HSub.hSub, Erdos107.f, Nat.choose, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem ersz_bounds :
    ∀ n ≥ 3, 2^(n - 2) + 1 ≤ f n ∧ f n ≤ Nat.choose (2 * n - 4) (n - 2) + 1 := by
…
```
Mentions: Erdos107.f
theorem Erdos107.variants.su_bound Suk [Su17] proved $$ f(n) ≤ 2^{(1+o(1))n}. $$ [Su17] Suk, Andrew, On the Erdős-Szekeres convex polygon problem sorry line 153 source
Source documentation
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.

depends on sorry what this means
Statement uses: Exists, Nat, Real, And, Asymptotics.IsLittleO, Filter.atTop, GE.ge, OfNat.ofNat, LE.le, Erdos107.f, HPow.hPow, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem su_bound :
    ∃ r : ℕ → ℝ, r =o[atTop] (fun n => (n : ℝ)) ∧
      ∀ n ≥ 3, (f n : ℝ) ≤ 2^(n + r n) := by
…
```
Mentions: Erdos107.f
theorem Erdos107.variants.hmpt_bound 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 sorry line 169 source
Source documentation
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.

depends on sorry what this means
Statement uses: Exists, Nat, Real, And, Asymptotics.IsBigO, Filter.atTop, Real.sqrt, HMul.hMul, Real.log, GE.ge, OfNat.ofNat, LE.le, Erdos107.f, HPow.hPow, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
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) := by
…
```
Mentions: Erdos107.f