  It is well known that there is a sharp density threshold for a random $r$-SAT
formula to be satisfiable, and a similar, smaller, threshold for it to be
satisfied by the pure literal rule. Also, above the satisfiability threshold,
where a random formula is with high probability (whp) unsatisfiable, the
unsatisfiability is whp due to a large "minimal unsatisfiable subformula"
(MUF).
  By contrast, we show that for the (rare) unsatisfiable formulae below the
pure literal threshold, the unsatisfiability is whp due to a unique MUF with
smallest possible "excess", failing this whp due to a unique MUF with the next
larger excess, and so forth. In the same regime, we give a precise asymptotic
expansion for the probability that a formula is unsatisfiable, and efficient
algorithms for satisfying a formula or proving its unsatisfiability. It remains
open what happens between the pure literal threshold and the satisfiability
threshold. We prove analogous results for the $k$-core and $k$-colorability
thresholds for a random graph, or more generally a random $r$-uniform
hypergraph.
