1027.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

3 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 1027

References:

erdosproblems.com/1027
[Er64e] Erdős, P., On a combinatorial problem. II. Acta Math. Acad. Sci. Hungar. (1964), 445-447.
[Er71] Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109.

Imports (1)

FormalConjecturesUtil

def Erdos1027.groundSet The ground set $X=\bigcup_{A\in\mathcal{F}}A$ of a finite family of finite sets line 40 source
Source documentation
The ground set $X=\bigcup_{A\in\mathcal{F}}A$ of a finite family of finite sets.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: DecidableEq, Finset
Proof uses: Finset.biUnion, id
```
def groundSet (F : Finset (Finset α)) : Finset α := F.biUnion id
…
```
def Erdos1027.goodSets The sets $B\subseteq X$ which intersect every set in $\mathcal{F}$, yet contain none of them line 44 source
Source documentation
The sets $B\subseteq X$ which intersect every set in $\mathcal{F}$, yet contain none of them.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: DecidableEq, Finset
Proof uses: Finset.filter, Membership.mem, And, Finset.Nonempty, Inter.inter, Not, LE.le, Finset.powerset, Erdos1027.groundSet
11 instances, coercions and auxiliary constants left out.
```
def goodSets (F : Finset (Finset α)) : Finset (Finset α) :=
…
```
theorem Erdos1027.erdos_1027 sorry line 61 source
depends on sorry what this means
Statement uses: Iff, True, Real, LT.lt, OfNat.ofNat, Exists, And, Filter.Eventually, Nat, DecidableEq, Finset, Membership.mem, Eq, Finset.card, LE.le, HMul.hMul, HPow.hPow, Erdos1027.groundSet, Erdos1027.goodSets, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.