FormalConjectures.ErdosProblems.«905»

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import FormalConjecturesUtil

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.

namespace Erdos905open Finset

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 This was proved independently by Edwards (unpublished) and Khadzhiivanov and Nikiforov [KhNi79].

@[category research solved, AMS 5, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos905.lean"]
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) := byn:ℕG:SimpleGraph (Fin n)inst✝:DecidableRel G.AdjhG:↑n ^ 2 / 4 < ↑(#G.edgeFinset)⊢ ∃ e ∈ G.edgeFinset, ↑n / 6 ≤ ↑(#(G.trianglesContaining e))
  sorryAll goals completed! 🐙

end Erdos905