The main theorem of this paper is: If ƒ is a partial function from 1 to 1 which is ∑11-bounded, then there is a weakly finite primitive recursive dilatorD such that for all infinite αεdom, ƒ D. The proof involves only elementary combinatorial constructions of trees. A generalization to ptykes is also given.

The main theorem of this paper is: If ƒ is a partial function from ℵ 1 to ℵ 1 which is ∑ 1 1 -bounded, then there is a weakly finite primitive recursive dilator D such that for all infinite αϵdom , ƒ ⩽ D . The proof involves only elementary combinatorial constructions of trees. A generalization to ptykes is also given.