1008.lean`

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

4 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 1008

References:

erdosproblems.com/1008
[CFS14b] Conlon, D. and Fox, J. and Sudakov, B., Large subgraphs without complete bipartite graphs. arXiv:1401.6711 (2014).
[Er71] Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109.

Imports (1)

FormalConjecturesUtil

theorem Erdos1008.erdos_1008 Does every graph with $m$ edges contain a subgraph with $\gg m^{2/3}$ edges which contains no $C_4$? sorry line 45 source
Source documentation
Does every graph with $m$ edges contain a subgraph with $\gg m^{2/3}$ edges which contains no $C_4$?
This problem was first solved in the affirmative by Conlon, Fox, and Sudakov [CFS14b]. A simple proof is given by Hunter in the comments.

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, GT.gt, OfNat.ofNat, Fintype, SimpleGraph, LE.le, SimpleGraph.Free, Fin, Nat, SimpleGraph.cycleGraph, HMul.hMul, HPow.hPow, Set.ncard, Sym2, SimpleGraph.edgeSet, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1008 : answer(True) ↔
    ∃ c > (0 : ℝ), ∀ (V : Type) [Fintype V] (G : SimpleGraph V),
      ∃ H ≤ G, (cycleGraph 4).Free H ∧
        c * (G.edgeSet.ncard : ℝ) ^ (2 / 3 : ℝ) ≤ (H.edgeSet.ncard : ℝ) := by
…
```
Mentions: c
theorem Erdos1008.erdos_1008.variants.three_quarters Originally asked by Bollobás and Erdős in 'a colloquium on graph theory at Tihany' with $m^{2/3}$ replaced by $m^{3/4}$ sorry line 57 source
Source documentation
Originally asked by Bollobás and Erdős in 'a colloquium on graph theory at Tihany' with $m^{2/3}$ replaced by $m^{3/4}$. Folkman showed this is false with the counterexample $K_{n,n^2}$, which has $n^3$ edges, and yet every subgraph with $>n^2+\binom{n}{2}$ edges contains a $C_4$.

depends on sorry what this means
Statement uses: Iff, False, Exists, Real, And, GT.gt, OfNat.ofNat, Fintype, SimpleGraph, LE.le, SimpleGraph.Free, Fin, Nat, SimpleGraph.cycleGraph, HMul.hMul, HPow.hPow, Set.ncard, Sym2, SimpleGraph.edgeSet, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1008.variants.three_quarters : answer(False) ↔
    ∃ c > (0 : ℝ), ∀ (V : Type) [Fintype V] (G : SimpleGraph V),
      ∃ H ≤ G, (cycleGraph 4).Free H ∧
        c * (G.edgeSet.ncard : ℝ) ^ (3 / 4 : ℝ) ≤ (H.edgeSet.ncard : ℝ) := by
…
```
Mentions: c
theorem Erdos1008.erdos_1008.variants.folkman Folkman's counterexample $K_{n,n^2}$, which has $n^3$ edges, and yet every subgraph with $>n^2+\binom{n}{2}$ edges contains a $C_4$ sorry line 68 source
Source documentation
Folkman's counterexample $K_{n,n^2}$, which has $n^3$ edges, and yet every subgraph with $>n^2+\binom{n}{2}$ edges contains a $C_4$.

depends on sorry what this means
Statement uses: Nat, And, Eq, Set.ncard, Sym2, Sum, Fin, HPow.hPow, OfNat.ofNat, SimpleGraph.edgeSet, completeBipartiteGraph, SimpleGraph, LE.le, LT.lt, HAdd.hAdd, Nat.choose, SimpleGraph.IsContained, SimpleGraph.cycleGraph
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1008.variants.folkman (n : ℕ) :
    ((completeBipartiteGraph (Fin n) (Fin (n ^ 2))).edgeSet.ncard = n ^ 3) ∧
      ∀ H ≤ completeBipartiteGraph (Fin n) (Fin (n ^ 2)),
        n ^ 2 + n.choose 2 < H.edgeSet.ncard → cycleGraph 4 ⊑ H := by
…
```
theorem Erdos1008.erdos_1008.variants.lower_bound In [Er71] Erdős revises the conjecture to $m^{2/3}$, and notes $\gg m^{1/2}$ is trivial sorry line 78 source
Source documentation
In [Er71] Erdős revises the conjecture to $m^{2/3}$, and notes $\gg m^{1/2}$ is trivial.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Fintype, SimpleGraph, LE.le, SimpleGraph.Free, Fin, Nat, SimpleGraph.cycleGraph, HMul.hMul, HPow.hPow, Set.ncard, Sym2, SimpleGraph.edgeSet, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1008.variants.lower_bound :
    ∃ c > (0 : ℝ), ∀ (V : Type) [Fintype V] (G : SimpleGraph V),
      ∃ H ≤ G, (cycleGraph 4).Free H ∧
        c * (G.edgeSet.ncard : ℝ) ^ (1 / 2 : ℝ) ≤ (H.edgeSet.ncard : ℝ) := by
…
```
Mentions: c