Indestructible colourings and rainbow Ramsey theorems

Lajos Soukup. "Indestructible colourings and rainbow Ramsey theorems." Fundamenta Mathematicae 202.2 (2009): 161-180. <http://eudml.org/doc/283334>.

@article{LajosSoukup2009,
abstract = {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 and such that $V^\{\}$ ⊨ f c.c.c.-indestructibly establishes $ω₁ ↛ *[(ω₁;ω₁)]_\{k-bdd\}$, but $V^\{\}$ ⊨ ω₁ is the union of countably many f-rainbow sets.},
author = {Lajos Soukup},
journal = {Fundamenta Mathematicae},
keywords = {anti Ramsey; rainbow Ramsey; polychromatic Ramsey; indestructible; forcing; partition relations; Martin's Axiom},
language = {eng},
number = {2},
pages = {161-180},
title = {Indestructible colourings and rainbow Ramsey theorems},
url = {http://eudml.org/doc/283334},
volume = {202},
year = {2009},
}

TY - JOUR
AU - Lajos Soukup
TI - Indestructible colourings and rainbow Ramsey theorems
JO - Fundamenta Mathematicae
PY - 2009
VL - 202
IS - 2
SP - 161
EP - 180
AB - 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 and such that $V^{}$ ⊨ f c.c.c.-indestructibly establishes $ω₁ ↛ *[(ω₁;ω₁)]_{k-bdd}$, but $V^{}$ ⊨ ω₁ is the union of countably many f-rainbow sets.
LA - eng
KW - anti Ramsey; rainbow Ramsey; polychromatic Ramsey; indestructible; forcing; partition relations; Martin's Axiom
UR - http://eudml.org/doc/283334
ER -