665.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

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 665

References:

erdosproblems.com/665
[Er97f] Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10.
[ErLa82] Erdős, P. and Larson, J., On pairwise balanced block designs with the sizes of blocks as uniform as possible. Annals of Discrete Mathematics (1982), 129-134.
[ShSi85] S. S. Shrikhande and N. M. Singhi, On a problem of Erdős and Larson. Combinatorica (1985), 351-358.

Imports (1)

FormalConjecturesUtil

theorem Erdos665.erdos_665 Documented declaration: erdos_665 sorry line 48 source
Source documentation
A pairwise balanced design for $\{1,\ldots,n\}$ is a collection of sets $A_1,\ldots,A_m\subseteq \{1,\ldots,n\}$ such that $2\leq \lvert A_i\rvert <n$ and every pair of distinct elements $x,y\in \{1,\ldots,n\}$ is contained in exactly one $A_i$. Is there a constant $C>0$ and, for all large $n$, a pairwise balanced design such that $$\lvert A_i\rvert > n^{1/2}-C$$ for all $1\leq i\leq m$?

L4DEL explanation
A pairwise balanced design for $\{1,\ldots,n\}$ is a collection of sets $A_1,\ldots,A_m\subseteq \{1,\ldots,n\}$ such that $2\leq \lvert A_i\rvert 0$ and, for all large $n$, a pairwise balanced design such that $$\lvert A_i\rvert > n^{1/2}-C$$ for all $1\leq i\leq m$?
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, Finset, Fin, Finset.IsPairwiseBalancedDesign, Membership.mem, Finset.card, HSub.hSub, Real.sqrt, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
23 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_665 :
    answer(sorry) ↔ ∃ C : ℝ, 0 < C ∧ ∀ᶠ n : ℕ in atTop,
    ∃ H : Finset (Finset (Fin n)), H.IsPairwiseBalancedDesign ∧
      ∀ e ∈ H, e.card < n ∧ Real.sqrt n - C < e.card := by
…
```