Factorization forests of finite height (Q908712)

We introduce factorization forests. It is shown that the vertex set of a dense factorization forest of finite height satisfies a bounded gap Ramsey-type property of words. The main result is that every morphism from a free semigroup to a finite semigroup S admits a Ramseyan factorization forest of height at most \(9| S|\). Technique for constructing factorization forests are developed.