903.lean`

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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 903

References:

erdosproblems.com/903
[EFSW85] Erdős, P. and Fowler, Joel C. and S'os, Vera T. and Wilson, Richard M., On {$2$}-designs. J. Combin. Theory Ser. A (1985), 131--142.
[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 Erdos903.erdos_903 Documented declaration: erdos_903 sorry line 45 source
Source documentation
Let $n=p^2+p+1$ for some prime power $p$, and let $A_1,\ldots,A_t\subseteq \{1,\ldots,n\}$ be a block design (so that every pair $x,y\in \{1,\ldots,n\}$ is contained in exactly one $A_i$). Is it true that if $t>n$ then $t\geq n+p$?
This is true, and was proved by Erdős, Fowler, Sós, and Wilson [EFSW85].

depends on sorry what this means
Statement uses: Iff, True, Nat, IsPrimePow, Finset, Fin, HAdd.hAdd, HPow.hPow, OfNat.ofNat, Finset.IsPairwiseBalancedDesign, LT.lt, Finset.card, LE.le
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_903 :
    answer(True) ↔ ∀ p : ℕ, IsPrimePow p →
    ∀ H : Finset (Finset (Fin (p ^ 2 + p + 1))),
      H.IsPairwiseBalancedDesign → p ^ 2 + p + 1 < H.card →
        p ^ 2 + p + 1 + p ≤ H.card := by
…
```