Structure of random r-SAT below the pure literal threshold

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.

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