734.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 734

References:

erdosproblems.com/734
[Er81] Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.
[dBEr48] de Bruijn, N. G. and Erdős, P., On a combinatorial problem. Nederl. Akad. Wetensch., Proc. (1948), 1277--1279 = Indagationes Math. 10, 421--423.

Imports (1)

FormalConjecturesUtil

theorem Erdos734.erdos_734 Find, for all large sorry line 43 source
Source documentation
Find, for all large $n$, a non-trivial pairwise balanced block design $A_1,\ldots,A_m\subseteq \{1,\ldots,n\}$ such that, for all $t$, there are $O(n^{1/2})$ many $i$ such that $\lvert A_i\rvert=t$.

depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, Finset, Fin, Finset.IsPairwiseBalancedDesign, Membership.mem, Finset.card, LE.le, Finset.filter, Eq, HMul.hMul, Real.sqrt, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
27 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_734 :
    ∃ C : ℝ, 0 < C ∧ ∀ᶠ n : ℕ in atTop,
    ∃ H : Finset (Finset (Fin n)), H.IsPairwiseBalancedDesign ∧
      (∀ e ∈ H, e.card < n) ∧ ∀ t : ℕ,
        ({e ∈ H | e.card = t}.card : ℝ) ≤ C * Real.sqrt n := by
…
```