We present an extension of Ramsey numbers, in particular the recently popular ordered Ramsey numbers, by considering graphs with a partial ordering on their vertices. In this context, we can use various families of posets in order to build host graphs for Ramsey problems, each having unique challenges. We focus mainly on Ramsey numbers arising from Boolean lattices and, in the 1-uniform case, explore connections to well-studied Turán problems. Beyond this, we find a strong difference between Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the partial-orderings on the graphs have large antichains.
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