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