505.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

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 505

Reference: erdosproblems.com/505

Borsuk's conjecture (1933): Is every bounded set of diameter 1 in $\mathbb{R}^n$ the union of at most $n + 1$ sets of diameter strictly less than 1?

Erdős [Er44] suspected this is false for sufficiently large $n$. Confirmed by Kahn–Kalai [KK93], who disproved the conjecture for $n \geq 2015$. The current best is $n \geq 63$ (Grinsztajn, 2026); the smallest refereed counterexample is $n = 64$ (Jenrich–Brouwer, 2014).

The conjecture is true for $n \leq 3$ (Eggleston [Eg55] for $n = 3$).

References

[Bo33] Borsuk, K. (1933). Drei Sätze über die n-dimensionale euklidische Sphäre. Fund. Math. 20, 177–190.
[Er44] Erdős, P. (1944). Remarks on a conjecture of Borsuk.
[Eg55] Eggleston, H. G. (1955). Covering a three-dimensional set with sets of smaller diameter. J. London Math. Soc. 30, 11–24.
[KK93] Kahn, J., Kalai, G. (1993). A counterexample to Borsuk's conjecture. Bull. Amer. Math. Soc. 29, 60–62.

This file points to the canonical formalization in FormalConjectures.Wikipedia.BorsukConjecture.

AI disclosure

Lean 4 code in this file was drafted with assistance from Claude (Anthropic). The mathematical content and references are the author's own work.

Imports (2)

theorem Erdos505.erdos_505 sorry line 72 source
depends on sorry what this means
Statement uses: Not, Nat, Borsuk.BorsukConjecture
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos505.erdos_505.small_dim Borsuk's conjecture, small dimensions: Borsuk.BorsukConjecture n holds for $n \leq 3$ sorry line 79 source
Source documentation
Borsuk's conjecture, small dimensions: Borsuk.BorsukConjecture n holds for $n \leq 3$. Elementary for $n \leq 1$, proved by Borsuk [Bo33] for $n = 2$ and by Eggleston [Eg55] for $n = 3$.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Borsuk.BorsukConjecture
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 erdos_505.small_dim (n : ℕ) (hn : n ≤ 3) : BorsukConjecture n := by
…
```