Towards finite homomorphism-homogeneous relational structures

Relational structure is homomorphism-homogeneous if every local homomorphism between finite induced substructures can be extended to endomorphism. The classification of homomorphism-homogeneous relational structures is still a challenging problem even for a finite case. In this work finite homomorphism-homogeneous binary relational structures having two relations that are both symmetric and irreflexive are classified. In addition to that, more general relational structures having finitely many relations of described type are considered. For those a classification is achieved when assuming that sets of colors assigned to pairs of vertices each one representing set of present edges between this pair constitute a linear partial order.