816.lean`

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

4 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 816

References:

erdosproblems.com/816
[Er91] Erdős, P., Problems and results in combinatorial analysis and combinatorial number theory. Graph theory, combinatorics, and applications, Vol. 1 (Kalamazoo, MI, 1988) (1991), 397-406.
[ChMa25] K. Chen and J. Ma, A problem of Erdős and Hajnal on paths with equal-degree endpoints. arXiv:2503.19569 (2025).

Imports (1)

FormalConjecturesUtil

def Erdos816.HasEqualDegreePathThree G contains two vertices of the same degree which are joined by a path of length 3 line 42 source
Source documentation
G contains two vertices of the same degree which are joined by a path of length 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, SimpleGraph
Proof uses: Exists, And, Ne, Eq, Nat, SimpleGraph.degree, Membership.mem, Set, SimpleGraph.neighborSet, Classical.propDecidable, SimpleGraph.Adj, SimpleGraph.Walk, SimpleGraph.Walk.IsPath, SimpleGraph.Walk.length, OfNat.ofNat
4 instances, coercions and auxiliary constants left out.
```
def HasEqualDegreePathThree {V : Type*} [Fintype V] (G : SimpleGraph V) : Prop :=
…
```
theorem Erdos816.erdos_816 sorry line 61 source
depends on sorry what this means
Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, SimpleGraph, Fin, HAdd.hAdd, HMul.hMul, Eq, Finset.card, Sym2, SimpleGraph.edgeFinset, Classical.propDecidable, SimpleGraph.Adj, HPow.hPow, Erdos816.HasEqualDegreePathThree
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos816.erdos_816.variants.chen_ma Chen and Ma [ChMa25] proved that, provided sorry line 72 source
Source documentation
Chen and Ma [ChMa25] proved that, provided $n\geq 600$, all graphs with $2n+1$ vertices and at least $n^2+n$ edges contain two vertices of the same degree joined by a path of length $3$, except $K_{n,n+1}$.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, SimpleGraph, Fin, HAdd.hAdd, HMul.hMul, HPow.hPow, Finset.card, Sym2, SimpleGraph.edgeFinset, Classical.propDecidable, SimpleGraph.Adj, Not, Nonempty, SimpleGraph.Iso, Sum, completeBipartiteGraph, Erdos816.HasEqualDegreePathThree
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_816.variants.chen_ma : ∀ n : ℕ, 600 ≤ n →
    ∀ G : SimpleGraph (Fin (2 * n + 1)), n ^ 2 + n ≤ G.edgeFinset.card →
      ¬ Nonempty (G ≃g completeBipartiteGraph (Fin n) (Fin (n + 1))) →
        HasEqualDegreePathThree G := by
…
```
Mentions: Erdos816.HasEqualDegreePathThree
theorem Erdos816.erdos_816.variants.complete_bipartite The example of $K_{n,n+1}$ shows that this fails if we only have $n^2+n$ edges sorry line 81 source
Source documentation
The example of $K_{n,n+1}$ shows that this fails if we only have $n^2+n$ edges.

depends on sorry what this means
Statement uses: Nat, And, Eq, Finset.card, Sym2, Sum, Fin, HAdd.hAdd, OfNat.ofNat, SimpleGraph.edgeFinset, completeBipartiteGraph, Classical.propDecidable, SimpleGraph.Adj, HPow.hPow, Not, Erdos816.HasEqualDegreePathThree
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_816.variants.complete_bipartite : ∀ n : ℕ,
    (completeBipartiteGraph (Fin n) (Fin (n + 1))).edgeFinset.card = n ^ 2 + n ∧
      ¬ HasEqualDegreePathThree (completeBipartiteGraph (Fin n) (Fin (n + 1))) := by
…
```
Mentions: Erdos816.HasEqualDegreePathThree