Rainbow Ramsey theorems for colorings establishing negative partition relations
Given a function $f$, a subset of its domain is a rainbow subset for $f$ if $f$ is one-to-one on it. We start with an old Erdős problem: Assume $f$ is a coloring of the pairs of $\omega _1$ with three colors such that every subset $ A $ of $\omega _1$ of size $\omega _1$ contains a pair of each color. Does there exist a rainbow triangle? We investigate rainbow problems and results of this style for colorings of pairs establishing negative “square bracket” relations.