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Interpolation percolation

Let X be a countably infinite set of real numbers and let Y_x, x \in X, be an independent family of stationary random subsets of the real numbers, e.g. homogeneous Poisson point processes. We give criteria for the a.s. existence of various "regular" functions f with the property that f(x) \in Y_x for all x \in X. Several open questions are posed.