Rainbow Ramsey theory at uncountable cardinals, part 1

A typical problem studied in Rainbow Ramsey theory is: given a coloring of pairs of ordinals such that each color is used “not too many times”, is it possible to find a subset that is “rainbow”, namely the coloring restricted to the pairs of this subset is injective. Different problems arise by varying the parameters, for example, the “type” of the source cardinal (successor of regular/regular, limit singular/regular) or the exact meaning of “not too many times”. In this talk, we will survey the known results on different variations, with some attention paid to the connection with other combinatorial principles in set theory (eg. saturation of ideals, stationary reflection, various square principles). The ultimate goal is to motivate unsolved questions in this area. The talk will be self-contained.