1075.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 1075

References:

erdosproblems.com/1075
[Er64f] Erdős, P., On extremal problems of graphs and generalized graphs. Israel J. Math. (1964), 183--190.

Imports (1)

FormalConjecturesUtil

theorem Erdos1075.erdos_1075 Let $r\geq 3$ sorry line 42 source
Source documentation
Let $r\geq 3$. There exists $c_r>r^{-r}$ such that, for any $\epsilon>0$, if $n$ is sufficiently large, the following holds. Any $r$-uniform hypergraph on $n$ vertices with at least $(1+\epsilon)(n/r)^r$ many edges contains a subgraph on $m$ vertices with at least $c_rm^r$ edges, where $m=m(n)\to \infty$ as $n\to \infty$.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Exists, Real, And, LT.lt, HPow.hPow, Inv.inv, Filter.Tendsto, Filter.atTop, Filter.Eventually, Finset, Fin, Finset.IsUniform, HMul.hMul, HAdd.hAdd, HDiv.hDiv, Finset.card, Eq and 1 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
48 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1075 :
    ∀ r : ℕ, 3 ≤ r → ∃ c : ℝ, (r : ℝ)⁻¹ ^ r < c ∧
    ∀ ε : ℝ, 0 < ε → ∃ m : ℕ → ℕ, Tendsto m atTop atTop ∧
      ∀ᶠ n : ℕ in atTop, ∀ H : Finset (Finset (Fin n)),
        H.IsUniform r → (1 + ε) * ((n : ℝ) / r) ^ r ≤ H.card →
          ∃ S : Finset (Fin n), S.card = m n ∧
            c * (S.card : ℝ) ^ r ≤ (H.hypergraphInduce S).card := by
…
```
Mentions: c