Uniformly hard languages

Ladner (1975) showed that there are no minimal recursive sets under polynomial-time reductions. Given any recursive set A, Ladner constructs a set B such that B strictly reduces to A but B does not lie in P. The set B does have very long sequences of input lengths of easily computable instances. We examine whether Ladner´s results hold if we restrict ourselves to “uniformly hard languages” which have no long sequences of easily computable instances. Under a hard to disprove assumption, we show that there exists a minimal recursive uniformly hard set under honest many-one polynomial-time reductions