767.lean`

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

5 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 767

References:

erdosproblems.com/767
[Er64c] Erdős, P., Extremal problems in graph theory. Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963) (1964), 29-36.
[Er69b] Erdős, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968) (1969), 27-35.
[Er75] Erdős, P., Some recent progress on extremal problems in graph theory. Congr. Numer. (1975), 3-14.
[Ji04] Jiang, Tao, A note on a conjecture about cycles with many incident chords. J. Graph Theory (2004), 180-182.

Imports (1)

FormalConjecturesUtil

def Erdos767.HasCycleWithIncidentChords G contains a cycle with k chords incident to a vertex on the cycle line 42 source
Source documentation
G contains a cycle with k chords incident to a vertex on the cycle.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Nat
Proof uses: Exists, SimpleGraph.Walk, And, SimpleGraph.Walk.IsCycle, Fin, Function.Injective, SimpleGraph.Walk.IsChord, Sym2.mk
```
def HasCycleWithIncidentChords {V : Type*} (G : SimpleGraph V) (k : ℕ) : Prop :=
…
```
def Erdos767.g g k n is the maximal number of edges possible on a graph with n vertices which does not contain a cycle with k chords incident to a vertex on the cycle line 49 source
Source documentation
g k n is the maximal number of edges possible on a graph with n vertices which does not contain a cycle with k chords incident to a vertex on the cycle.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, Exists, SimpleGraph, Fin, And, Not, Erdos767.HasCycleWithIncidentChords, Eq, Finset.card, Sym2, SimpleGraph.edgeFinset, Classical.propDecidable, SimpleGraph.Adj
4 instances, coercions and auxiliary constants left out.
```
noncomputable def g (k n : ℕ) : ℕ :=
…
```
theorem Erdos767.erdos_767 sorry line 63 source
depends on sorry what this means
Statement uses: Iff, True, Nat, GE.ge, OfNat.ofNat, HAdd.hAdd, HMul.hMul, Eq, Erdos767.g, HSub.hSub, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos767.erdos_767.variants.czipszer Czipszer proved that $g_k(n)\leq (k+1)n$ sorry line 69 source
Source documentation
Czipszer proved that $g_k(n)\leq (k+1)n$.

depends on sorry what this means
Statement uses: Nat, LE.le, Erdos767.g, HMul.hMul, HAdd.hAdd, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_767.variants.czipszer (k n : ℕ) : g k n ≤ (k + 1) * n := by
…
```
Mentions: Erdos767.g
theorem Erdos767.erdos_767.variants.posa Pósa proved that $g_1(n)=2n-4$ for $n\geq 4$ sorry line 74 source
Source documentation
Pósa proved that $g_1(n)=2n-4$ for $n\geq 4$.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Eq, Erdos767.g, HSub.hSub, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_767.variants.posa (n : ℕ) (hn : 4 ≤ n) : g 1 n = 2 * n - 4 := by
…
```
Mentions: Erdos767.g