The smallest regular polytopes of given rank

An abstract polytope is called regular   if its automorphism group has a single orbit on flags (maximal chains). In this paper, the regular nn-polytopes with the smallest number of flags are found, for every rank n>1n>1. With a few small exceptions, the smallest regular nn-polytopes come from a family of ‘tight’ polytopes with 2⋅4n−12⋅4n−1 flags, one for each nn, with Schläfli symbol {4∣4∣⋯∣4}{4∣4∣⋯∣4}. Also with few exceptions, these have both the smallest number of elements, and the smallest number of edges in their Hasse diagram.