Abstract
We prove equivalences of asymmetric partition relations involving natural numbers and products of weakly compact cardinals ?, infinite cardinals ?<? 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(? 22,3), r(??2,3) and r(??3,3). We moreover provide general upper bounds for r(? 2 m,3) and r(??m,n).
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