In this talk, we consider the possibility that for every avoidable pattern P , there exists a finite set S of forbidden patterns and factors, containg P , such that the words over Σ_λ(P ) are essentially the factors of an HDOL word. This is a strong version of Cassaigne's conjecture. I will give many examples of such HDOL words characterized by forbidden patterns and factors, as well as related open problems. We will also discuss the factor complexity of words avoiding patterns.