A causal set is a poset which is reflexive, antisymmetric, transitive and locally finite.
As a motivation, there is a programme to model spacetime as fundamentally discrete, with causal sets providing the underlying structure. Typically this is constructed by sprinkling (a poisson process) existing spacetime with elements, endowing these elements with an ordering given by causal cones, and removing spacetime. Volume is then given by a counting metric, which with the causal structure is enough to build a geometry. By the poisson nature of this process, the distribution is Lorentz-invariant.
This only makes sense in nature if the causal set is manifold-like, by which we mean it can be faithfully embedded into a manifold, such that the element count gives volume.
Precisely when is a causal set manifold-like: what are the necessary conditions for the existence of such an embedding? (Are there interesting sufficient conditions?) Do they have natural interpretations?
[This should be tagged quantum-gravity I think.]