Continuous colorings associated with certain characteristics of the continuum

A coloring c:[X]n→Z is said to be irreducible if for every Y⊆X of equal cardinality c″[Y]n=Z. The focus of this note will be to show that there are continuous irreducible colorings on sets of reals associated with various cardinal invariants of the continuum. It is interesting that some of the colorings make crucial use of exponential lower bounds which have been proven for a certain class of finite Ramsey numbers.