747.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 747

References:

erdosproblems.com/747
[JKV08] Johansson, Anders and Kahn, Jeff and Vu, Van, Factors in random graphs. Random Structures Algorithms (2008), 1-28.
[Ka23] Kahn, Jeff, Asymptotics for Shamir's problem. Adv. Math. (2023), Paper No. 109019, 39.

Imports (1)

FormalConjecturesUtil

theorem Erdos747.erdos_747 How large should $\ell(n)$ be such that, almost surely, a random $3$-uniform hypergraph on $3n$ vertices with $\ell(n)$ edges must contain $n$ vertex-disjoint edges? sorry line 45 source
Source documentation
How large should $\ell(n)$ be such that, almost surely, a random $3$-uniform hypergraph on $3n$ vertices with $\ell(n)$ edges must contain $n$ vertex-disjoint edges?
The precise asymptotic was given by Kahn [Ka23], proving that the threshold is $\sim n\log n$ (also for the general problem over $r$-uniform hypergraphs).

depends on sorry what this means
Statement uses: Nat, Real, HMul.hMul, Real.log, LT.lt, OfNat.ofNat, And, Filter.Tendsto, Hypergraph.matchingProbability, Nat.floor, HAdd.hAdd, Filter.atTop, nhds, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
50 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_747 :
    let threshold : ℕ → ℝ := answer(fun n : ℕ ↦ n * Real.log n)
…
```