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 >>>

We give a quantitative combinatorial characterization of size-oblivious one-sided testability in the dense graph model, resolving a question of Alon, Fischer, Newman, and Shapira. For hereditary graph properties, we prove that one-sided testability is quantitatively equivalent to the existence of suitable hypergraph containers, a central and widely used tool in ... more >>>

A Boolean-weighted graph is a finite graph whose edges carry DNF formulas over a common variable set. This paper studies two isomorphism problems on such graphs, distinguished by whether the per-edge condition requires the matched edge labels to be syntactically DNF-isomorphic (BWG-ISO) or syntactically DNF non-isomorphic (BWG-NI), each over a ... more >>>