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 show how two techniques from statistical physics can be adapted to solve a variant of the notorious Unique Games problem, potentially opening new avenues towards the Unique Games Conjecture. The variant, which we call Count Unique Games, is a promise problem in which the "yes" case guarantees a certain number of highly satisfiable assignments to the Unique Games instance. In the standard Unique Games problem, the "yes" case only guarantees at least one such assignment. We exhibit efficient algorithms for Count Unique Games based on approximating a suitable partition function for the Unique Games instance via (i) a zero-free region and polynomial interpolation, and (ii) the cluster expansion. We also show that a modest improvement to the parameters for which we give results would be strong negative evidence for the truth of the Unique Games Conjecture.

A famous theorem of Dirac states that any graph on $n$ vertices with minimum degree at least $n/2$ has a Hamilton cycle. Such graphs are called Dirac graphs. Strengthening this result, we show the existence of rainbow Hamilton cycles in $\mu n$-bounded colorings of Dirac graphs for sufficiently small $\mu >0$.