Abstract. Let [Formula: see text] and [Formula: see text] be randomly chosen subsets of the first [Formula: see text] positive integers of cardinalities [Formula: see text], such that their sumset [Formula: see text] has size [Formula: see text]. We show that asymptotically almost surely [Formula: see text] and [Formula: see text] are almost fully contained in arithmetic progressions [Formula: see text] and [Formula: see text] with the same common difference and cardinalities approximately [Formula: see text]. We also prove a counting theorem for such pairs of sets in arbitrary abelian groups. The results hold for [Formula: see text] and [Formula: see text]. Our main tool is an asymmetric version of the method of hypergraph containers which was recently used by Campos to prove similar results in the special case [Formula: see text].

We consider one-sided error property testing of $\mathcal{F}$-minor freeness in bounded-degree graphs for any finite family of graphs $\mathcal{F}$ that contains a minor of $K_{2,k}$, the $k$-circus graph, or the $(k\times 2)$-grid for any $k\in\mathbb{N}$. This includes, for instance, testing whether a graph is outerplanar or a cactus graph. The query complexity of our algorithm in terms of the number of vertices in the graph, $n$, is $\tilde{O}(n^{2/3} / ε^5)$. Czumaj et~al.\ showed that cycle-freeness and $C_k$-minor freeness can be tested with query complexity $\tilde{O}(\sqrt{n})$ by using random walks, and that testing $H$-minor freeness for any $H$ that contains a cycles requires $Ω(\sqrt{n})$ queries. In contrast to these results, we analyze the structure of the graph and show that either we can find a subgraph of sublinear size that includes the forbidden minor $H$, or we can find a pair of disjoint subsets of vertices whose edge-cut is large, which induces an $H$-minor.