extremalNumber_le_iff_of_nonneg — Mathlib

English

If m is nonnegative, then the same extremalNumber ≤ m characterization holds for the floor of m in a suitable setting.

Русский

Если $m\\ge 0$, то та же характеристика \$\\mathrm{extremalNumber} \\le m\$ верна для целочисленного снижения модуля.

LaTeX

$$\$ extremalNumber( card V) H \\le m \\iff \\forall G, H.Free G \\Rightarrow |G.edgeFinset| ≤ m \$$$

Lean4

@[inherit_doc extremalNumber_le_iff]
theorem extremalNumber_le_iff_of_nonneg (H : SimpleGraph W) {m : R} (h : 0 ≤ m) :
    extremalNumber (card V) H ≤ m ↔ ∀ ⦃G : SimpleGraph V⦄ [DecidableRel G.Adj], H.Free G → #G.edgeFinset ≤ m :=
  by
  simp_rw [← Nat.le_floor_iff h]
  exact extremalNumber_le_iff H ⌊m⌋₊

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