  Kim defined a very general combinatorial abstraction of the diameter of
polytopes called subset partition graphs to study how certain combinatorial
properties of such graphs may be achieved in lower bound constructions. Using
Lov\'asz' Local Lemma, we give a general randomized construction for subset
partition graphs satisfying strong adjacency and end-point count properties.
This can be used as a building block to conceptually simplify the constructions
given in [Kim11].
  We also use our method to construct abstract spindles, an analogy to the
spindles used by Santos to disprove the Hirsch conjecture, of exponential
length which satisfy the adjacency and end-point count properties.
