207.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 207

References:

erdosproblems.com/207
[KSSS22b] Kwan, M. and Sah, A. and Sawhney, M. and Simkin, M., High-girth Steiner triple systems. arXiv:2201.04554 (2022).

Imports (1)

FormalConjecturesUtil

theorem Erdos207.erdos_207 Documented declaration: erdos_207 sorry line 47 source
Source documentation
For any $g\geq 2$, if $n$ is sufficiently large and $\equiv 1,3\pmod{6}$ then there exists a 3-uniform hypergraph on $n$ vertices such that
- every pair of vertices is contained in exactly one edge (i.e. the graph is a Steiner triple system) and
- for any $2\leq j\leq g$ any collection of $j$ edges contains at least $j+3$ vertices.
Proved by Kwan, Sah, Sawhney, and Simkin [KSSS22b].
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Filter.Eventually, Or, Eq, HMod.hMod, Exists, Finset, Fin, And, Finset.IsBlockDesign, Finset.card, HAdd.hAdd, Finset.biUnion, id, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_207 :
    ∀ g : ℕ, 2 ≤ g → ∀ᶠ n : ℕ in atTop,
    n % 6 = 1 ∨ n % 6 = 3 → ∃ H : Finset (Finset (Fin n)),
      H.IsBlockDesign 2 3 ∧ ∀ E ⊆ H,
        2 ≤ E.card → E.card ≤ g → E.card + 3 ≤ (E.biUnion id).card := by
…
```