Finite homomorphism dualities
Jan Foniok, AlCo, Lix, Ecole Polytechnique

A finite homomorphism duality is a pair (F,D) of finite sets of relational
structures that satisfy: whenever g is a structure, either there exists
f in F and a homomorphism from f to g, or there exists d in D and a
homomorphism from g to d, but not both.

I will show a full characterisation of finite homomorphism dualities
and some of their (somewhat unexpected) appearances in the ordering of
structures by existence of a homomorphism (gaps & maximal antichains),
and constraint satisfaction problems (first order definability).