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import FormalConjecturesUtilErdős Problem 590
[Ch72] Chang, C. C., A partition theorem for the complete graph on {$\omega\sp{\omega }$}. J. Combinatorial Theory Ser. A (1972), 396-452.
[Sp57] Specker, Ernst, Teilmengen von Mengen mit Relationen. Comment. Math. Helv. (1957), 302-314.
[La73] Larson, Jean A., A short proof of a partition theorem for the ordinal {$\omega \sp{\omega }$}. Ann. Math. Logic (1973/74), 129-145.
open Cardinal Ordinal
namespace Erdos590
universe u
Let $α$ be the infinite ordinal $\omega^{\omega}$. It was proved by Chang [Ch72] that any red/blue colouring of the edges of $K_α$ there is either a red $K_α$ or a blue $K_3$.
@[category research solved, AMS 3]
theorem erdos_590 : OrdinalCardinalRamsey (ω ^ ω) (ω ^ ω) 3 := ⊢ OrdinalCardinalRamsey (ω ^ ω) (ω ^ ω) 3
All goals completed! 🐙
Specker [Sp57] proved that when $α=ω^2$ any red/blue colouring of the edges of $K_α$ there is either a red $K_α$ or a blue $K_3$.
@[category research solved, AMS 3]
theorem erdos_590.variants.two : OrdinalCardinalRamsey (ω ^ 2) (ω ^ 2) 3 := ⊢ OrdinalCardinalRamsey (ω ^ 2) (ω ^ 2) 3
All goals completed! 🐙
Specker [Sp57] proved that when $α=ω^n$ for $3≤ n < \omega$ then it is not the case that any red/blue colouring of the edges of $K_α$ there is either a red $K_α$ or a blue $K_3$.
@[category research solved, AMS 3]
theorem erdos_590.variants.ge_three_false {n : ℕ} (h : 3 ≤ n) :
¬ OrdinalCardinalRamsey (ω ^ n) (ω ^ n) 3 := n:ℕh:3 ≤ n⊢ ¬OrdinalCardinalRamsey (ω ^ n) (ω ^ n) 3
All goals completed! 🐙
Let m be a finite cardinal $< \omega$. Let $α$ be the infinite ordinal $\omega^{\omega}$. It was proved by Milnor that any red/blue colouring of the edges of $K_α$ there is either a red $K_α$ or a blue $K_3$. A shorter proof was found by Larson [La73]
@[category research solved, AMS 3]
theorem erdos_590.variants.finite_cardinal (m : ℕ) : OrdinalCardinalRamsey (ω ^ ω) (ω ^ ω) m := m:ℕ⊢ OrdinalCardinalRamsey (ω ^ ω) (ω ^ ω) ↑m
All goals completed! 🐙
end Erdos590