Rainbow Ramsey simple structures

A relational structure R is rainbow Ramsey   if for every finite induced substructure C of R and every colouring of the copies of C with countably many colours, such that each colour is used at most kk times for a fixed kk, there exists a copy R∗ of R so that the copies of C in R∗ use each colour at most once.We show that a class of homogeneous binary relational structures generalizing the Rado graph are rainbow Ramsey. Via compactness this then implies that for all finite graphs B and C and k∈ωk∈ω, there exists a graph A so that for every colouring of the copies of C in A such that each colour is used at most kk times, there exists a copy B∗ of B in A so that the copies of C in B∗ use each colour at most once.