Idiosynchromatic Poetry

Abstract

We prove equivalences of asymmetric partition relations involving natural numbers and products of weakly compact cardinals $κ$, infinite cardinals $λ<κ$ and natural numbers to certain classes of finitary problems in the theory of edge-coloured digraphs. We are able to determine three classes of Ramsey numbers exactly, typical examples in these classes are $r(ω^2 2, 3)$, $r(κλ2, 3)$ and $r(κλ3, 3)$. We moreover provide general upper bounds for $r(ω^2 m, 3)$ and $r(κλm, n)$.