On uniquely partitionable relational structures and object systems

We introduce object systems as a common generalization of graphs, hypergraphs, digraphs and relational structures. Let C be a concrete category, a simple object system over C is an ordered pair S = (V, E), ∗Research supported in part by Slovak VEGA Grant 2/4134/24. 282 J. Bucko and P. Mihók where E = {A1, A2, . . . , Am} is a finite set of the objects of C, such that the ground-set V (Ai) of each object Ai ∈ E is a finite set with at least two elements and V ⊇ mi=1 V (Ai). To generalize the results on graph colourings to simple object systems we define, analogously as for graphs, that an additive induced-hereditary property of simple object systems over a category C is any class of systems closed under isomorphism, induced-subsystems and disjoint union of systems, respectively. We present a survey of recent results and conditions for object systems to be uniquely partitionable into subsystems of given properties.