1079.lean`

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

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 1079

References:

erdosproblems.com/1079
[Er75] Erdős, P., Some recent progress on extremal problems in graph theory. Congr. Numer. (1975), 3-14.
[BoTh81] Bollobás, Béla and Thomason, Andrew, Dense neighbourhoods and Turán's theorem. J. Combin. Theory Ser. B (1981), 111--114.
[Bo83b] Bondy, J. A., Large dense neighbourhoods and Turán's theorem. J. Combin. Theory Ser. B (1983), 109--111.

Imports (1)

FormalConjecturesUtil

theorem Erdos1079.erdos_1079 Let $r\geq 4$ sorry line 55 source
Source documentation
Let $r\geq 4$. If $G$ is a graph on $n$ vertices with at least $\mathrm{ex}(n;K_r)$ edges then must $G$ contain a vertex with degree $d\gg_r n$ whose neighbourhood contains at least $\mathrm{ex}(d;K_{r-1})$ edges?
As Erdős [Er75] says 'if true this would be a nice generalisation of Turán's theorem'. This is true (unless $G$ it itself the Turán graph), and was proved by Bollobás and Thomason [BoTh81]. Bondy [Bo83b] showed that if $G$ has $>\mathrm{ex}(n;K_r)$ edges then the corresponding vertex can be chosen to be of maximum degree in $G$.
The number of edges in the neighbourhood of $v$ is the number of edges of $G$ both of whose endpoints are adjacent to $v$. Graphs on a single vertex are excluded, since there every degree is $0$.

depends on sorry what this means
Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, Exists, Real, And, LT.lt, SimpleGraph, Fin, SimpleGraph.extremalNumber, SimpleGraph.completeGraph, Set.ncard, Sym2, SimpleGraph.edgeSet, HMul.hMul, SimpleGraph.degree, Membership.mem, Set and 5 more
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_1079 : answer(True) ↔ ∀ r : ℕ, 4 ≤ r → ∃ c : ℝ, 0 < c ∧ ∀ n : ℕ, 2 ≤ n →
    ∀ G : SimpleGraph (Fin n), extremalNumber n (completeGraph (Fin r)) ≤ G.edgeSet.ncard →
      ∃ v : Fin n, c * n ≤ G.degree v ∧
        extremalNumber (G.degree v) (completeGraph (Fin (r - 1))) ≤
          {e ∈ G.edgeSet | ∀ x ∈ e, G.Adj v x}.ncard := by
…
```
Mentions: c
theorem Erdos1079.erdos_1079.variants.bondy sorry line 69 source
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, SimpleGraph, Fin, DecidableRel, SimpleGraph.Adj, LT.lt, SimpleGraph.extremalNumber, SimpleGraph.completeGraph, Set.ncard, Sym2, SimpleGraph.edgeSet, Exists, And, Eq, SimpleGraph.degree, Membership.mem, Set, SimpleGraph.neighborSet and 4 more
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.