74.lean`

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

8 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 74

Reference: erdosproblems.com/74

Imports (1)

FormalConjecturesUtil

def Erdos74.SimpleGraph.edgeDistancesToBipartite For a given subgraph A, this is the set of all numbers k such that A can be made bipartite by deleting k edges line 43 source
Source documentation
For a given subgraph A, this is the set of all numbers k such that A can be made bipartite by deleting k edges.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, SimpleGraph.Subgraph, Set, Nat
Proof uses: Set.ofPred, Exists, Sym2, LE.le, SimpleGraph.Subgraph.edgeSet, SimpleGraph.IsBipartite, SimpleGraph.Subgraph.verts, SimpleGraph.Subgraph.deleteEdges, SimpleGraph.Subgraph.coe, Eq, Set.ncard
2 instances, coercions and auxiliary constants left out.
```
def SimpleGraph.edgeDistancesToBipartite {G : SimpleGraph V} (A : G.Subgraph) : Set ℕ :=
…
```
theorem Erdos74.SimpleGraph.edgeDistancesToBipartite_nonempty The set of edge distances to a bipartite graph is always non-empty because deleting all edges from a graph makes it bipartite line 52 source
Source documentation
The set of edge distances to a bipartite graph is always non-empty because deleting all edges from a graph makes it bipartite.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, SimpleGraph.Subgraph, Set.Nonempty, Nat, Erdos74.SimpleGraph.edgeDistancesToBipartite
Proof uses: id, Exists.intro, Membership.mem, Set, Set.ofPred, Exists, Sym2, LE.le, SimpleGraph.Subgraph.edgeSet, SimpleGraph.IsBipartite, SimpleGraph.Subgraph.verts, SimpleGraph.Subgraph.deleteEdges, SimpleGraph.Subgraph.coe, Eq, Set.ncard, Nonempty.intro, SimpleGraph.Coloring, Fin, OfNat.ofNat, RelHom.mk and 17 more
16 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
theorem SimpleGraph.edgeDistancesToBipartite_nonempty {G : SimpleGraph V} (A : G.Subgraph) :
    SimpleGraph.edgeDistancesToBipartite A |>.Nonempty := by
…
```
Mentions: Erdos74.SimpleGraph.edgeDistancesToBipartite
def Erdos74.SimpleGraph.minEdgeDistToBipartite The minimum number of edges that must be deleted from a subgraph A to make it bipartite line 62 source
Source documentation
The minimum number of edges that must be deleted from a subgraph A to make it bipartite.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, SimpleGraph.Subgraph, Nat
Proof uses: InfSet.sInf, Erdos74.SimpleGraph.edgeDistancesToBipartite
1 instances, coercions and auxiliary constants left out.
```
noncomputable def SimpleGraph.minEdgeDistToBipartite {G : SimpleGraph V} (A : G.Subgraph) : ℕ :=
…
```
def Erdos74.SimpleGraph.subgraphEdgeDistsToBipartite For a graph G and a number n, this is the set of minEdgeDistToBipartite A for all induced subgraphs A of G on n vertices line 69 source
Source documentation
For a graph G and a number n, this is the set of minEdgeDistToBipartite A for all induced subgraphs A of G on n vertices.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Nat, Set
Proof uses: Set.ofPred, Exists, SimpleGraph.Subgraph, Eq, Set.ncard, SimpleGraph.Subgraph.verts, Set.Finite, Erdos74.SimpleGraph.minEdgeDistToBipartite
```
def SimpleGraph.subgraphEdgeDistsToBipartite (G : SimpleGraph V) (n : ℕ) : Set ℕ :=
…
```
theorem Erdos74.SimpleGraph.subgraphEdgeDistsToBipartite_bddAbove The set of minimum edge distances to bipartite for subgraphs of size n is bounded above line 80 source
Source documentation
The set of minimum edge distances to bipartite for subgraphs of size n is bounded above. A graph on n vertices has at most n choose 2 edges, and deleting all of them makes the graph bipartite, providing a straightforward upper bound.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Nat, BddAbove, Erdos74.SimpleGraph.subgraphEdgeDistsToBipartite
Proof uses: Exists.intro, Membership.mem, Set, upperBounds, Nat.choose, OfNat.ofNat, Eq.mpr, Exists, SimpleGraph.Subgraph, Eq, Set.ncard, SimpleGraph.Subgraph.verts, Set.Finite, InfSet.sInf, Set.ofPred, Sym2, LE.le, SimpleGraph.Subgraph.edgeSet, SimpleGraph.IsBipartite, SimpleGraph.Subgraph.deleteEdges and 66 more
35 instances, coercions and auxiliary constants left out.

Tactic trace: 23 steps
Open to load the checked before-and-after goal states.
```
theorem SimpleGraph.subgraphEdgeDistsToBipartite_bddAbove (G : SimpleGraph V) (n : ℕ) :
    BddAbove (SimpleGraph.subgraphEdgeDistsToBipartite G n) := by
…
```
Mentions: Erdos74.SimpleGraph.subgraphEdgeDistsToBipartite
def Erdos74.SimpleGraph.maxSubgraphEdgeDistToBipartite For a given graph $G$ and size $n$, this defines the smallest number $k$ such that any subgraph of $G$ on $n$ vertices can be made bipartite by deleting at most $k$ edges line 118 source
Source documentation
For a given graph $G$ and size $n$, this defines the smallest number $k$ such that any subgraph of $G$ on $n$ vertices can be made bipartite by deleting at most $k$ edges.
This value is optimal because it is the maximum of minEdgeDistToBipartite taken over all $n$-vertex subgraphs. This means there exists at least one $n$-vertex subgraph that requires exactly this many edge deletions. This is Definition 3.1 in [EHS82].
[EHS82] Erdős, P. and Hajnal, A. and Szemerédi, E., On almost bipartite large chromatic graphs Theory and practice of combinatorics (1982), 117-123.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Nat
Proof uses: SupSet.sSup, Erdos74.SimpleGraph.subgraphEdgeDistsToBipartite
1 instances, coercions and auxiliary constants left out.
```
noncomputable def SimpleGraph.maxSubgraphEdgeDistToBipartite
    (G : SimpleGraph V) (n : ℕ) : ℕ := sSup <| SimpleGraph.subgraphEdgeDistsToBipartite G n
…
```
Mentions: Erdos74.SimpleGraph.subgraphEdgeDistsToBipartite
theorem Erdos74.erdos_74 sorry line 131 source
depends on sorry what this means
Statement uses: Iff, False, Nat, Filter.Tendsto, Filter.atTop, Exists, SimpleGraph, And, Eq, ENat, SimpleGraph.chromaticNumber, Top.top, LE.le, Erdos74.SimpleGraph.maxSubgraphEdgeDistToBipartite
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
7 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos74.erdos_74.variants.sqrt Is there a graph of infinite chromatic number such that every finite subgraph on $n$ vertices can be made bipartite by deleting at most $\sqrt{n}$ edges? sorry line 141 source
Source documentation
Is there a graph of infinite chromatic number such that every finite subgraph on $n$ vertices can be made bipartite by deleting at most $\sqrt{n}$ edges?

depends on sorry what this means
Statement uses: Iff, True, Exists, SimpleGraph, And, Eq, ENat, SimpleGraph.chromaticNumber, Top.top, Nat, LE.le, Real, Erdos74.SimpleGraph.maxSubgraphEdgeDistToBipartite, Real.sqrt
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_74.variants.sqrt : answer(sorry) ↔
    ∃ (V : Type u) (G : SimpleGraph V), G.chromaticNumber = ⊤ ∧
    ∀ n, G.maxSubgraphEdgeDistToBipartite n ≤ (n : ℝ).sqrt := by
…
```