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 >>>

We extend the bounded degree graph model for property testing introduced by Goldreich and Ron (Algorithmica, 2002) to hypergraphs. In this framework, we analyse the query complexity of three fundamental hypergraph properties: colorability, $k$-partiteness, and independence number. We present a randomized algorithm for testing $k$-partiteness within families of $k$-uniform $n$-vertex ... more >>>

We prove the first hardness results against efficient proof search by quantum algorithms. We show that under Learning with Errors (LWE), the standard lattice-based cryptographic assumption, no quantum algorithm can weakly automate $\mathbf{TC}^0$-Frege. This extends the line of results of Kraí?ek and Pudlák (Information and Computation, 1998), Bonet, Pitassi, and ... more >>>