Indestructible colourings and rainbow Ramsey theorems

• Autorzy: Lajos Soukup
• Języki publikacji: EN

Abstrakty

EN

We show that if a colouring c establishes ω₂ ↛ [(ω₁:ω)]² then c establishes this negative partition relation in each Cohen-generic extension of the ground model, i.e. this property of c is Cohen-indestructible. This result yields a negative answer to a question of Erdős and Hajnal: it is consistent that GCH holds and there is a colouring c:[ω₂]² → 2 establishing ω₂ ↛ [(ω₁:ω)]₂ such that some colouring g:[ω₁]² → 2 does not embed into c.
It is also consistent that $2^{ω₁}$ is arbitrarily large, and there is a function g establishing $2^{ω₁} ↛ [(ω₁,ω₂)]_{ω₁}$; but there is no uncountable g-rainbow subset of $2^{ω₁}$.
We also show that if GCH holds then for each k ∈ ω there is a k-bounded colouring f: [ω₁]² → ω₁ and there are two c.c.c. posets 𝓟 and 𝒬 such that
$V^{𝓟}$ ⊨ f c.c.c.-indestructibly establishes $ω₁ ↛ *[(ω₁;ω₁)]_{k-bdd}$,
but
$V^{𝒬}$ ⊨ ω₁ is the union of countably many f-rainbow sets.

Kategorie tematyczne

• 03E50: Continuum hypothesis and Martin's axiom
• 03E02: Partition relations
• 05D10: Ramsey theory
• 03E35: Consistency and independence results

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2009
• Tom: 202
• Numer: 2
• Strony: 161-180

Daty

wydano

2009

Twórcy

autor

Lajos Soukup

• Alfréd Rényi Institute of Mathematics, V. Reáltanoda utca 13-15, H-1053 Budapest, Hungary

Identyfikatory

DOI

10.4064/fm202-2-4