Topological dynamics of automorphism groups, ultrafilter combinatorics, and the Generic Point Problem

For G a closed subgroup of S∞, we provide a precise combinatorial characterization of when the universal minimal flow M(G) is metrizable. In particular, each such instance fits into the framework of metrizable flows developed in [KPT] and [NVT]; as a consequence, each G with metrizable universal minimal flow has the generic point property, i.e. every minimal G-flow has a point whose orbit is comeager. This solves the Generic Point Problem raised in [AKL] for closed subgroups of S∞.