1067.lean`

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

3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 1067

References:

erdosproblems.com/1067
[BoPi24] N. Bowler and M. Pitz, A note on uncountably chromatic graphs. arXiv:2402.05984 (2024).
[ErHa66] Erdős, P. and Hajnal, A., On chromatic number of graphs and set-systems. Acta Math. Acad. Sci. Hungar. (1966), 61-99.
[Ko13] Komjáth, Péter, A note on chromatic number and connectivity of infinite graphs. Israel J. Math. (2013), 499--506.
[So15] Soukup, Dániel T., Trees, ladders and graphs. J. Combin. Theory Ser. B (2015), 96--116.
[Th17] Thomassen, Carsten, Infinitely connected subgraphs in graphs of uncountable chromatic number. Combinatorica (2017), 785--793.

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100SolvedSet1.lean

def Erdos1067.InfinitelyEdgeConnected A graph is infinitely edge-connected if to disconnect the graph requires deleting infinitely many edges line 46 source
Source documentation
A graph is infinitely edge-connected if to disconnect the graph requires deleting infinitely many edges. In other words, removing any finite set of edges leaves the graph connected.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph
Proof uses: Set, Sym2, Set.Finite, SimpleGraph.Connected, SimpleGraph.deleteEdges
```
def InfinitelyEdgeConnected {V : Type*} (G : SimpleGraph V) : Prop :=
…
```
theorem Erdos1067.erdos_1067 Does every graph with chromatic number $\aleph_1$ contain an infinitely connected subgraph with chromatic number $\aleph_1$? sorry line 60 source
Source documentation
Does every graph with chromatic number $\aleph_1$ contain an infinitely connected subgraph with chromatic number $\aleph_1$?
Komjáth [Ko13] proved that it is consistent that the answer is no. This was improved by Soukup [So15], who constructed a counterexample using no extra set-theoretical assumptions. A simpler elementary example was given by Bowler and Pitz [BoPi24].
This was formalized in Lean by Alexeev using Aristotle and Aleph Prover.

depends on sorry what this means
Statement uses: Iff, False, SimpleGraph, Eq, Cardinal, SimpleGraph.chromaticCardinal, OrderEmbedding, Ordinal, Cardinal.aleph, OfNat.ofNat, Exists, SimpleGraph.Subgraph, And, SimpleGraph.Subgraph.verts, SimpleGraph.Subgraph.coe, SimpleGraph.InfinitelyConnected
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
19 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1067 :
    answer(False) ↔ ∀ (V : Type) (G : SimpleGraph V), G.chromaticCardinal = ℵ_ 1 →
      ∃ (H : G.Subgraph), H.coe.chromaticCardinal = ℵ_ 1 ∧ InfinitelyConnected H.coe := by
…
```
theorem Erdos1067.erdos_1067.variants.infinite_edge_connectivity Does every graph with chromatic number $\aleph_1$ contain an infinitely edge-connected subgraph with chromatic number $\aleph_1$? sorry line 73 source
Source documentation
Does every graph with chromatic number $\aleph_1$ contain an infinitely edge-connected subgraph with chromatic number $\aleph_1$?
Thomassen [Th17] proved that the answer is yes: every graph of uncountable chromatic number has a subgraph with infinite edge-connectivity and uncountable chromatic number.

depends on sorry what this means
Statement uses: Iff, True, SimpleGraph, Eq, Cardinal, SimpleGraph.chromaticCardinal, OrderEmbedding, Ordinal, Cardinal.aleph, OfNat.ofNat, Exists, SimpleGraph.Subgraph, And, SimpleGraph.Subgraph.verts, SimpleGraph.Subgraph.coe, Erdos1067.InfinitelyEdgeConnected
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
19 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1067.variants.infinite_edge_connectivity :
    answer(True) ↔ ∀ (V : Type) (G : SimpleGraph V), G.chromaticCardinal = ℵ_ 1 →
      ∃ (H : G.Subgraph), H.coe.chromaticCardinal = ℵ_ 1 ∧ InfinitelyEdgeConnected H.coe := by
…
```
Mentions: Erdos1067.InfinitelyEdgeConnected