579.lean`

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4 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 579

References:

erdosproblems.com/579
[EHSS83] P. Erdős, A. Hajnal, V. T. Sós and E. Szemerédi, More results on Ramsey–Turán type problems, Combinatorica 3 (1983), 69–81.

Imports (1)

FormalConjecturesUtil

abbrev Erdos579.octahedron The octahedron $K_{2,2,2}$: the complete tripartite graph with all three parts of size $2$ line 37 source
Source documentation
The octahedron $K_{2,2,2}$: the complete tripartite graph with all three parts of size $2$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Sigma, Fin, OfNat.ofNat, Nat
Proof uses: SimpleGraph.completeMultipartiteGraph
2 instances, coercions and auxiliary constants left out.
```
abbrev octahedron : SimpleGraph (Σ _ : Fin 3, Fin 2) :=
…
```
theorem Erdos579.erdos_579 Let $\delta > 0$ sorry line 55 source
Source documentation
Let $\delta > 0$. If $n$ is sufficiently large and $G$ is a graph on $n$ vertices with no $K_{2,2,2}$ (the octahedron) and at least $\delta n^2$ edges, must $G$ contain an independent set of size $\gg_\delta n$?
This is a problem of Erdős, Hajnal, Sós, and Szemerédi [EHSS83]. It is open; they proved the statement for $\delta > 1/8$ (see erdos_579.variants.ehss_large_delta), and the difficulty is to push the edge-density threshold down to an arbitrary $\delta > 0$.
Here $K_{2,2,2}$ is the complete tripartite graph with all parts of size $2$, encoded as completeMultipartiteGraph (fun _ : Fin 3 => Fin 2); "contains no $K_{2,2,2}$" is expressed via SimpleGraph.Free.

depends on sorry what this means
Statement uses: Iff, True, Real, LT.lt, OfNat.ofNat, Exists, And, Filter.Eventually, Nat, SimpleGraph, Fin, SimpleGraph.Free, Sigma, Erdos579.octahedron, LE.le, HMul.hMul, HPow.hPow, Finset.card, Sym2, SimpleGraph.edgeFinset and 4 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_579 : answer(sorry) ↔
    ∀ δ : ℝ, 0 < δ → ∃ c : ℝ, 0 < c ∧ ∀ᶠ n : ℕ in atTop,
      ∀ G : SimpleGraph (Fin n), octahedron.Free G →
        δ * (n : ℝ) ^ 2 ≤ G.edgeFinset.card →
          c * n ≤ (G.indepNum : ℝ) := by
…
```
Mentions: c
theorem Erdos579.erdos_579.variants.ehss_large_delta The partial result of Erdős, Hajnal, Sós, and Szemerédi [EHSS83]: the statement of erdos_579 holds whenever the edge-density coefficient exceeds $1/8$ sorry line 71 source
Source documentation
The partial result of Erdős, Hajnal, Sós, and Szemerédi [EHSS83]: the statement of erdos_579 holds whenever the edge-density coefficient exceeds $1/8$. That is, for every $\delta > 1/8$ there is a $c > 0$ such that for all sufficiently large $n$, every $K_{2,2,2}$-free graph $G$ on $n$ vertices with at least $\delta n^2$ edges has an independent set of size at least $c n$.

depends on sorry what this means
Statement uses: Real, LT.lt, HDiv.hDiv, OfNat.ofNat, Nat, Exists, And, Filter.Eventually, SimpleGraph, Fin, SimpleGraph.Free, Sigma, Erdos579.octahedron, LE.le, HMul.hMul, HPow.hPow, Finset.card, Sym2, SimpleGraph.edgeFinset, Classical.propDecidable and 3 more
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_579.variants.ehss_large_delta (δ : ℝ) (hδ : 1 / 8 < δ) :
    ∃ c : ℝ, 0 < c ∧ ∀ᶠ n : ℕ in atTop,
      ∀ G : SimpleGraph (Fin n), octahedron.Free G →
        δ * (n : ℝ) ^ 2 ≤ G.edgeFinset.card →
          c * n ≤ (G.indepNum : ℝ) := by
…
```
Mentions: c
theorem Erdos579.erdos_579.variants.octahedron_not_free Sanity check that the forbidden structure is non-trivial: the octahedron is of course not octahedron-free, since it contains a copy of itself line 83 source
Source documentation
Sanity check that the forbidden structure is non-trivial: the octahedron is of course not octahedron-free, since it contains a copy of itself.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Not, SimpleGraph.Free, Sigma, Fin, OfNat.ofNat, Nat, Erdos579.octahedron
Proof uses: SimpleGraph.IsContained.rfl
2 instances, coercions and auxiliary constants left out.
```
theorem erdos_579.variants.octahedron_not_free : ¬ octahedron.Free octahedron :=
…
```
Mentions: Erdos579.octahedron