127.lean`

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L4DEL: Lean4 Domain Explanation Language

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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

4 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 127

References:

erdosproblems.com/127
[Er97b] Erdős, Paul, Some old and new problems in various branches of combinatorics. Discrete Math. (1997), 227-231.
[Ed73] Edwards, C. S., Some extremal properties of bipartite subgraphs. Canadian J. Math. (1973), 475-485.
[Al96] Alon, Noga, Bipartite subgraphs. Combinatorica (1996), 301-311.

Imports (1)

FormalConjecturesUtil

def Erdos127.f f m is the largest k such that every graph with m edges contains a bipartite subgraph with at least $\frac{m}{2}+\frac{\sqrt{8m+1}-1}{8}+k$ edges line 40 source
Source documentation
f m is the largest k such that every graph with m edges contains a bipartite subgraph with at least $\frac{m}{2}+\frac{\sqrt{8m+1}-1}{8}+k$ edges.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, Fintype, SimpleGraph, Eq, Set.ncard, Sym2, SimpleGraph.edgeSet, Exists, And, LE.le, SimpleGraph.IsBipartite, Real, HAdd.hAdd, HDiv.hDiv, OfNat.ofNat, HSub.hSub, Real.sqrt, HMul.hMul
19 instances, coercions and auxiliary constants left out.
```
noncomputable def f (m : ℕ) : ℕ :=
…
```
theorem Erdos127.erdos_127 sorry line 57 source
depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, And, Filter.Tendsto, Filter.atTop, Erdos127.f
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos127.erdos_127.variants.edwards sorry line 65 source
depends on sorry what this means
Statement uses: Finite, SimpleGraph, Exists, And, LE.le, SimpleGraph.IsBipartite, Real, HAdd.hAdd, HDiv.hDiv, Set.ncard, Sym2, SimpleGraph.edgeSet, OfNat.ofNat, Nat, HSub.hSub, Real.sqrt, HMul.hMul
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.
theorem Erdos127.erdos_127.variants.alon_upper Alon [Al96] showed that $f(m)\ll m^{1/4}$ for all $m$ sorry line 72 source
Source documentation
Alon [Al96] showed that $f(m)\ll m^{1/4}$ for all $m$.

depends on sorry what this means
Statement uses: Exists, Real, Nat, LE.le, Erdos127.f, HMul.hMul, HPow.hPow, HDiv.hDiv, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_127.variants.alon_upper : ∃ C : ℝ, ∀ m : ℕ, (f m : ℝ) ≤ C * (m : ℝ) ^ (1 / 4 : ℝ) := by
…
```
Mentions: Erdos127.f