We discuss a new direction in which the use of some methods from arithmetic combinatorics can be extended. We consider functions taking values in Euclidean space and supported on subsets of {1, 2, . . ., N}. In this context we present a proof of a natural generalization of Szemerédi's theorem. We also prove a similar generalization of a theorem of Sárkőzy using a vector-valued Fourier transform, adapting an argument of Green and obtaining effective bounds.

For k prime and A a finite set of integers with |A| ≥ 3(k − 1)2(k − 1)! we prove that |A + k · A| ≥ (k + 1)|A| − ⌈k(k + 2)/4⌉ where k · A = {ka: a ∈ A}. We also describe the sets for which equality holds.

Belief propagation (BP) is a message-passing algorithm that computes the exact marginal distributions at every vertex of a graphical model without cycles. While BP is designed to work correctly on trees, it is routinely applied to general graphical models that may contain cycles, in which case neither convergence, nor correctness in the case of convergence is guaranteed. Nonetheless, BP has gained popularity as it seems to remain effective in many cases of interest, even when the underlying graph is ‘far’ from being a tree. However, the theoretical understanding of BP (and its new relative survey propagation) when applied to CSPs is poor.

Contributing to the rigorous understanding of BP, in this paper we relate the convergence of BP to spectral properties of the graph. This encompasses a result for random graphs with a ‘planted’ solution; thus, we obtain the first rigorous result on BP for graph colouring in the case of a complex graphical structure (as opposed to trees). In particular, the analysis shows how belief propagation breaks the symmetry between the 3! possible permutations of the colour classes.

We prove that there exists a constant c such that, for any integer Δ, the Ramsey number of a bipartite graph on n vertices with maximum degree Δ is less than 2cΔn. A probabilistic argument due to Graham, Rödl and Ruciński implies that this result is essentially sharp, up to the constant c in the exponent. Our proof hinges upon a quantitative form of a hypergraph packing result of Rödl, Ruciński and Taraz.

We model a problem about networks built from wireless devices using identifying and locating–dominating codes in unit disk graphs. It is known that minimizing the size of an identifying code is -complete even for bipartite graphs. First, we improve this result by showing that the problem remains -complete for bipartite planar unit disk graphs. Then, we address the question of the existence of an identifying code for random unit disk graphs. We derive the probability that there exists an identifying code as a function of the radius of the disks, and we find that for all interesting ranges of r this probability is bounded away from one. The results obtained are in sharp contrast to those concerning random graphs in the Erdős–Rényi model. Another well-studied class of codes is that of locating–dominating codes, which are less demanding than identifying codes. A locating–dominating code always exists, but minimizing its size is still -complete in general. We extend this result to our setting by showing that this question remains -complete for arbitrary planar unit disk graphs. Finally, we study the minimum size of such a code in random unit disk graphs, and we prove that with probability tending to one, it is of size (n/r)2/3+o(1) if r ≤ /2−ϵ is chosen such that nr2 → ∞, and of size n1+o(1) if nr2 ≪ lnn.

We show that the complete list of regular excluded minors for the class of signed-graphic matroids is M*(G1),. . . , M*(G29), R15, R16. Here G1,. . . , G29 are the vertically 2-connected excluded minors for the class of projective-planar graphs and R15 and R16 are two regular matroids that we will define in the article.

Stein's method of exchangeable pairs is examined through five examples in relation to Poisson and normal distribution approximation. In particular, in the case where the exchangeable pair is constructed from a reversible Markov chain, we analyse how modifying the step size of the chain in a natural way affects the error term in the approximation acquired through Stein's method. It has been noted for the normal approximation that smaller step sizes may yield better bounds, and we obtain the first rigorous results that verify this intuition. For the examples associated to the normal distribution, the bound on the error is expressed in terms of the spectrum of the underlying chain, a characteristic of the chain related to convergence rates. The Poisson approximation using exchangeable pairs is less studied than the normal, but in the examples presented here the same principles hold.