Regular families of forests, antichains and duality pairs of relational   structures
  Homomorphism duality pairs play crucial role in the theory of relational structures and in the Constraint Satisfaction Problem. The case where both classes are finite is fully characterized. The case when both side are infinite seems to be very complex. It is also known that no finite-infinite duality pair is possible if we make the additional restriction that both classes are antichains. In this paper we characterize the infinite-finite antichain dualities and infinite-finite dualities with trees or forest on the left hand side. This work builds on our earlier papers that gave several examples of infinite-finite antichain duality pairs of directed graphs and a complete characterization for caterpillar dualities. 
P\'eter L. Erd\H{o}s, D\"om\"ot\"or P\'alv\"olgyi, Claude Tardif,   G\'abor Tardos
math.CO cs.DM