On Unavoidable Sets of Word Patterns

We introduce the notion of unavoidable (complete) sets of word patterns, which is a refinement for that of words, and study certain numerical characteristics for unavoidable sets of patterns. In some cases we employ the graph of pattern overlaps introduced in this paper, which is a subgraph of the de Bruijn graph and which we prove to be Hamiltonian. In other cases we reduce a problem under consideration to known facts on unavoidable sets of words. We also give a relation between our problem and the intensively studied universal cycles, and prove that there exists a universal cycle for word patterns of any length over any alphabet. The Stirling numbers of the second kind and the Möbius function appear in our results.