Undecidability in ω-Regular Languages

In the infinite Post Correspondence Problem an instance (h, g) consists of two morphisms h and g, and the problem is to determine whether or not there exists an infinite word α such that h(α) = g(α). In the general case this problem was shown to be undecidable by K. Ruohonen (1985). Recently, it was proved that the infinite PCP is undecidable already when the domain alphabet of the morphisms consists of at least 9 letters. Here we show that the problem is undecidable for instances where the morphisms have a domain of 6 letters, when we restrict to solutions of ω-languages of the form R where R is a given regular language.