Erdős Problem 590

References:

• erdosproblems.com/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.

theorem Erdos590.erdos_590 : OrdinalCardinalRamsey (Ordinal.omega0 ^ Ordinal.omega0) (Ordinal.omega0 ^ Ordinal.omega0) 3

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 $K_3$.

theorem Erdos590.erdos_590.variants.two : OrdinalCardinalRamsey (Ordinal.omega0 ^ 2) (Ordinal.omega0 ^ 2) 3

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 $K_3$.

theorem Erdos590.erdos_590.variants.ge_three_false {n : ℕ} (h : 3 ≤ n) : ¬OrdinalCardinalRamsey (Ordinal.omega0 ^ n) (Ordinal.omega0 ^ n) 3

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 $K_3$.

theorem Erdos590.erdos_590.variants.finite_cardinal (m : ℕ) : OrdinalCardinalRamsey (Ordinal.omega0 ^ Ordinal.omega0) (Ordinal.omega0 ^ Ordinal.omega0) ↑m

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]