addRothNumber_le_ruzsaSzemerediNumber

English

Equality of nat-based RS-number with the standard RS-number via finite type card.

Русский

Справедливость между Nat-Card RS и стандартным RS через кардинал конечного типа.

LaTeX

$$$$ ruzsaSzemerediNumberNat_card : ruzsaSzemerediNumberNat (card α) = ruzsaSzemerediNumber α. $$$$

Lean4

theorem addRothNumber_le_ruzsaSzemerediNumber :
    card α * addRothNumber (univ : Finset α) ≤ ruzsaSzemerediNumber (Sum α (Sum α α)) :=
  by
  obtain ⟨s, -, hscard, hs⟩ := addRothNumber_spec (univ : Finset α)
  haveI := noAccidental hs
  rw [← hscard, ← card_triangleIndices, ← card_triangles]
  exact (locallyLinear hs).le_ruzsaSzemerediNumber

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