Unary functions

We consider F the class of finite unary functions, B the class of finite bijections and F k , k 1 , the class of finite k - 1 functions. We calculate Ramsey degrees for structures in F and F k , 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 M n F , n 1 , of the class F which contains finite structures of the form ( A , f 1 , . . . , f n ) where each f i 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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