Unary functions

We consider FF the class of finite unary functions, BB the class of finite bijections and FkFk, k>1k>1, the class of finite k−1k−1 functions. We calculate Ramsey degrees for structures in FF and FkFk, and we show that BB is a Ramsey class. We prove Ramsey property for the class OFOF which contains structures of the form (A,f,≤)(A,f,≤) where (A,f)∈F(A,f)∈F and ≤is a linear ordering on the set AA. We also consider a generalization MnFMnF, n>1n>1, of the class FF which contains finite structures of the form (A,f1,...,fn)(A,f1,...,fn) where each fifi is a unary function on the set AA. Finally we give a topological interpretation of our results by expanding the list of extremely amenable groups and by calculating various universal minimal flows.