$k$-SAT is one of the best known among a wide class of random
constraint satisfaction problems believed to exhibit a threshold
phenomenon where the control parameter is the ratio, number of
constraints to number of variables. There has been a large amount of
work towards estimating ... more >>>

We prove a general upper bound for scaling windows of sparse monotone covering problems. From that, we deduce that for every fixed value $k\geq 3$, the window of random $k$-SAT is $O(n/\log n)$, improving the Friedgut-Bourgain bound of $O(n/\log\log n)$. We also show that random signed Not-All-Equal-$k$-SAT and hypergraph non-two-colourability ... more >>>