1207.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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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

2 declarations, 1 contain sorry, 0 declared as axioms

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.
[LPZ26] Lee, S., Pohoata, C., and Zhu, D. G., The Minkowski grid has robustly many repeated distances. arXiv:2607.05374 (2026).
[PaTa02] Pach, János and Tardos, Gábor, Isosceles triangles determined by a planar point set. Graphs Combin. (2002), 769--779.

Imports (1)

FormalConjecturesUtil

def Erdos1207.P 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 $m$ line 46 source
Source documentation
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 $m$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: InfSet.sInf, Set.ofPred, Exists, Finset, EuclideanSpace, Real, Fin, And, Eq, Finset.card, SupSet.sSup, LE.le, Set.IsIsoscelesFree, OfNat.ofNat, ENNReal
16 instances, coercions and auxiliary constants left out.
```
noncomputable def P (d n : ℕ) : ℕ :=
…
```
theorem Erdos1207.erdos_1207 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 sorry line 57 source
Source documentation
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 $c>0$?
Lee, Pohoata, and Zhu [LPZ26] proved the stated planar bound. Their isosceles-free subsets also exclude degenerate triangles of equally spaced collinear points.

L4DEL explanation
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)0$? Lee, Pohoata, and Zhu [LPZ26] proved the stated planar bound. Their isosceles-free subsets also exclude degenerate triangles of equally spaced collinear points.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, GT.gt, OfNat.ofNat, Filter.Eventually, Nat, LT.lt, Erdos1207.P, HPow.hPow, HSub.hSub, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1207 : answer(True) ↔
    ∃ c > (0 : ℝ), ∀ᶠ n : ℕ in atTop, (P 2 n : ℝ) < (n : ℝ) ^ (1 - c) := by
…
```
Mentions: c, Erdos1207.P