Finite versus infinite: An insufficient shift

The shift graph GS is defined on the space of infinite subsets of natural numbers by letting two sets be adjacent if one can be obtained from the other by removing its least element. We show that this graph is not a minimum among the graphs of the form Gf defined on some Polish space X, where two distinct points are adjacent if one can be obtained from the other by a given Borel function f:X→X. This answers the primary outstanding question from [8].