905.lean`

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

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 905

References:

erdosproblems.com/905
[KhNi79] Khadzhiivanov, N. G. and Nikiforov, S. V., Solution of a problem of P. Erdős about the maximum number of triangles with a common edge in a graph. C. R. Acad. Bulgare Sci. (1979), 1315-1318.

Imports (1)

FormalConjecturesUtil

theorem Erdos905.erdos_905 Every graph with $n$ vertices and $>n^2/4$ edges contains an edge which is in at least $n/6$ triangles sorry line 44 source
Source documentation
Every graph with $n$ vertices and $>n^2/4$ edges contains an edge which is in at least $n/6$ triangles.
A conjecture of Bollobás and Erdős. This was proved independently by Edwards (unpublished) and Khadzhiivanov and Nikiforov [KhNi79].

depends on sorry what this means
Statement uses: Nat, SimpleGraph, Fin, DecidableRel, SimpleGraph.Adj, LT.lt, Real, HDiv.hDiv, HPow.hPow, OfNat.ofNat, Finset.card, Sym2, SimpleGraph.edgeFinset, Exists, And, Membership.mem, Finset, LE.le, SimpleGraph.trianglesContaining
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
40 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_905 (n : ℕ) (G : SimpleGraph (Fin n)) [DecidableRel G.Adj]
    (hG : (n : ℝ) ^ 2 / 4 < (#G.edgeFinset : ℝ)) :
    ∃ e ∈ G.edgeFinset, (n : ℝ) / 6 ≤ (#(G.trianglesContaining e) : ℝ) := by
…
```