2008/04/29 by Soukup, Lajos
#03E02 #03E35 #03E50 #05D10 #Combinatorics (math.CO) #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.0804.4548
We give a negative answer to a question of Erdos and Hajnal: it is consistent that GCH holds and there is a colouring c:[ω2]2→ 2 establishing ω2 \not→ [(ω1;ω)]22 such that some colouring g:[ω1]2→ 2 can not be embedded into c. It is also consistent that 2ω1 is arbitrarily large, and a function g establishes 2ω1 \not→ [(ω1,ω2)]2ω1 such that there is no uncountable g-rainbow subset of 2ω1. We also show that for each k∈ ω it is consistent with Martin's Axiom that the negative partition relation ω1 \not→^* [(ω1;ω1)]k-bdd holds.