In this paper, we consider the so-called “Furstenberg set problem” in high dimensions. First, following Wolff’s work on the two-dimensional real case, we provide “reasonable” upper bounds for the problem for $\mathbb{R}$ or $\mathbb{F}_{p}$. Next we study the “critical” case and improve the “trivial” exponent by ${\rm\Omega}(1/n^{2})$ for $\mathbb{F}_{p}^{n}$. Our key tool in obtaining this lower bound is a theorem about how things behave when the Loomis–Whitney inequality is nearly sharp, as it helps us to reduce the problem to dimension two.

Over 50 years ago, Erdős and Gallai conjectured that the edges of every graph on n vertices can be decomposed into O(n) cycles and edges. Among other results, Conlon, Fox and Sudakov recently proved that this holds for the random graph G(n, p) with probability approaching 1 as n → ∞. In this paper we show that for most edge probabilities G(n, p) can be decomposed into a union of n/4 + np/2 + o(n) cycles and edges w.h.p. This result is asymptotically tight.

We define a pseudometric on the set of all unbounded subsets of a metric space. The Kolmogorov quotient of this pseudometric space is a complete metric space. The definition of the pseudometric is guided by the principle that two unbounded subsets have distance 0 whenever they stay sublinearly close. Based on this pseudometric we introduce and study a general concept of boundaries of metric spaces. Such a boundary is the closure of a subset in the Kolmogorov quotient determined by an arbitrarily chosen family of unbounded subsets. Our interest lies in those boundaries which we get by choosing unbounded cyclic sub(semi)groups of a finitely generated group (or more general of a compactly generated, locally compact Hausdorff group). We show that these boundaries are quasi-isometric invariants and determine them in the case of nilpotent groups as a disjoint union of certain spheres (or projective spaces). In addition we apply this concept to vertex-transitive graphs with polynomial growth and to random walks on nilpotent groups.

In this paper we give a short proof of the Random Ramsey Theorem of Rödl and Ruciński: for any graph F which contains a cycle and r ≥ 2, there exist constants c, C > 0 such that

$$\begin{equation*}\Pr[G_{n,p} \rightarrow (F)_r^e] =\begin{cases}1-o(1) &p\ge Cn^{-1/m_2(F)},\\o(1) &p\le cn^{-1/m_2(F)},\end{cases}\end{equation*}$$

$$\begin{equation*}m_2(F) = \max_{J\subseteq F, v_J\ge 2} \frac{e_J-1}{v_J-2}.\end{equation*}$$

We show that for every sufficiently large n, the number of monotone subsequences of length four in a permutation on n points is at least

\begin{equation*}\binom{\lfloor{n/3}\rfloor}{4} + \binom{\lfloor{(n+1)/3}\rfloor}{4} + \binom{\lfloor{(n+2)/3}\rfloor}{4}.\end{equation*}

We provide an introduction to enumerating and constructing invariants of group representations via character methods. The problem is contextualized via two case studies, arising from our recent work: entanglement invariants for characterizing the structure of state spaces for composite quantum systems; and Markov invariants, a robust alternative to parameter-estimation intensive methods of statistical inference in molecular phylogenetics.

We establish a connection between epidemic models on random networks with general infection times considered in Barbour and Reinert (2013) and first passage percolation. Using techniques developed in Bhamidi, van der Hofstad and Hooghiemstra (2012), when each vertex has infinite contagious periods, we extend results on the epidemic curve in Barbour and Reinert (2013) from bounded degree graphs to general sparse random graphs with degrees having finite second moments as n → ∞, with an appropriate X 2log+ X condition. We also study the epidemic trail between the source and typical vertices in the graph.

Motivated by the analysis of social networks, we study a model of random networks that has both a given degree distribution and a tunable clustering coefficient. We consider two types of growth process on these graphs that model the spread of new ideas, technologies, viruses, or worms: the diffusion model and the symmetric threshold model. For both models, we characterize conditions under which global cascades are possible and compute their size explicitly, as a function of the degree distribution and the clustering coefficient. Our results are applied to regular or power-law graphs with exponential cutoff and shed new light on the impact of clustering.

Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in d-space, with distance parameter r and intensities λ and μ. We show for d ≥ 2 that if λ is supercritical for the one-type random geometric graph with distance parameter 2r, there exists μ such that (λ, μ) is supercritical (this was previously known for d = 2). For d = 2, we also consider the restriction of this graph to points in the unit square. Taking μ = τ λ for fixed τ, we give a strong law of large numbers as λ → ∞ for the connectivity threshold of this graph.

The probability h(n, m) that the block counting process of the Bolthausen-Sznitman n-coalescent ever visits the state m is analyzed. It is shown that the asymptotic hitting probabilities h(m) = limn→∞ h(n, m), m ∈ N, exist and an integral formula for h(m) is provided. The proof is based on generating functions and exploits a certain convolution property of the Bolthausen-Sznitman coalescent. It follows that h(m) ∼ 1/log m as m → ∞. An application to linear recursions is indicated.

We consider a continuum percolation model consisting of two types of nodes, namely legitimate and eavesdropper nodes, distributed according to independent Poisson point processes in R 2 of intensities λ and λE , respectively. A directed edge from one legitimate node A to another legitimate node B exists provided that the strength of the signal transmitted from node A that is received at node B is higher than that received at any eavesdropper node. The strength of the signal received at a node from a legitimate node depends not only on the distance between these nodes, but also on the location of the other legitimate nodes and an interference suppression parameter γ. The graph is said to percolate when there exists an infinitely connected component. We show that for any finite intensity λE of eavesdropper nodes, there exists a critical intensity λc < ∞ such that for all λ > λc the graph percolates for sufficiently small values of the interference parameter. Furthermore, for the subcritical regime, we show that there exists a λ0 such that for all λ < λ0 ≤ λc a suitable graph defined over eavesdropper node connections percolates that precludes percolation in the graphs formed by the legitimate nodes.

Consider a random multigraph G * with given vertex degrees d 1,…, d n , constructed by the configuration model. We give a new proof of the fact that, asymptotically for a sequence of such multigraphs with the number of edges the probability that the multigraph is simple stays away from 0 if and only if The new proof uses the method of moments, which makes it possible to use it in some applications concerning convergence in distribution. Corresponding results for bipartite graphs are included.

Let ${\rm\Gamma}(n,p)$ denote the binomial model of a random triangular group. We show that there exist constants $c,C>0$ such that if $p\leqslant c/n^{2}$, then asymptotically almost surely (a.a.s.) ${\rm\Gamma}(n,p)$ is free, and if $p\geqslant C\log n/n^{2}$, then a.a.s. ${\rm\Gamma}(n,p)$ has Kazhdan’s property (T). Furthermore, we show that there exist constants $C^{\prime },c^{\prime }>0$ such that if $C^{\prime }/n^{2}\leqslant p\leqslant c^{\prime }\log n/n^{2}$, then a.a.s. ${\rm\Gamma}(n,p)$ is neither free nor has Kazhdan’s property (T).

In this paper, we study a finite connected graph which admits a quasi-monomorphism to hyperbolic spaces and give a geometric bound for the Cheeger constants in terms of the volume, an upper bound of the degree, and the quasi-monomorphism.

One way to study certain classes of polynomials is by considering examples that are attached to combinatorial objects. Any graph $G$ has an associated reciprocal polynomial $R_{G}$ , and with two particular classes of reciprocal polynomials in mind one can ask the questions: (a) when is $R_{G}$ a product of cyclotomic polynomials (giving the cyclotomic graphs)? (b) when does $R_{G}$ have the minimal polynomial of a Salem number as its only non-cyclotomic factor (the non-trivial Salem graphs)? Cyclotomic graphs were classified by Smith (Combinatorial structures and their applications, Proceedings of Calgary International Conference, Calgary, AB, 1969 (eds R. Guy, H. Hanani, H. Saver and J. Schönheim; Gordon and Breach, New York, 1970) 403–406); the maximal connected ones are known as Smith graphs. Salem graphs are ‘spectrally close’ to being cyclotomic, in that nearly all their eigenvalues are in the critical interval $[-2,2]$ . On the other hand, Salem graphs do not need to be ‘combinatorially close’ to being cyclotomic: the largest cyclotomic induced subgraph might be comparatively tiny.

We define an $m$ -Salem graph to be a connected Salem graph $G$ for which $m$ is minimal such that there exists an induced cyclotomic subgraph of $G$ that has $m$ fewer vertices than $G$ . The $1$ -Salem subgraphs are both spectrally close and combinatorially close to being cyclotomic. Moreover, every Salem graph contains a $1$ -Salem graph as an induced subgraph, so these $1$ -Salem graphs provide some necessary substructure of all Salem graphs. The main result of this paper is a complete combinatorial description of all $1$ -Salem graphs: in the non-bipartite case there are $25$ infinite families and $383$ sporadic examples.

By using row convex tableaux, we study the section rings of Bott–Samelson varieties of type A. We obtain flat deformations and standard monomial type bases of the section rings. In a separate section, we investigate a three-dimensional Bott–Samelson variety in detail and compute its Hilbert polynomial and toric degenerations.

Let $Q$ be a finite quiver without oriented cycles, and let $k$ be an algebraically closed field. The main result in this paper is that there is a natural bijection between the elements in the associated Weyl group $W_{Q}$ and the cofinite additive quotient closed subcategories of the category of finite dimensional right modules over $kQ$. We prove this correspondence by linking these subcategories to certain ideals in the preprojective algebra associated to $Q$, which are also indexed by elements of $W_{Q}$.

Let $G$ be a graph of order $n\geq 6$ with minimum degree ${\it\delta}(G)\geq 4$. Arkin and Hassin [‘Graph partitions with minimum degree constraints’, Discrete Math. 190 (1998), 55–65] conjectured that there exists a bipartition $S,T$ of $V(G)$ such that $\lfloor n/2\rfloor -2\leq |S|,|T|\leq \lceil n/2\rceil +2$ and the minimum degrees in the subgraphs induced by $S$ and $T$ are at least two. In this paper, we first show that $G$ has a bipartition such that the minimum degree in each part is at least two, and then prove that the conjecture is true if the complement of $G$ contains no complete bipartite graph $K_{3,r}$, where $r=\lfloor n/2\rfloor -3$.

For any positive integer $n$, let $f(n)$ denote the number of 1-shell totally symmetric plane partitions of $n$. Recently, Hirschhorn and Sellers [‘Arithmetic properties of 1-shell totally symmetric plane partitions’, Bull. Aust. Math. Soc.89 (2014), 473–478] and Yao [‘New infinite families of congruences modulo 4 and 8 for 1-shell totally symmetric plane partitions’, Bull. Aust. Math. Soc.90 (2014), 37–46] proved a number of congruences satisfied by $f(n)$. In particular, Hirschhorn and Sellers proved that $f(10n+5)\equiv 0\ (\text{mod}\ 5)$. In this paper, we establish the generating function of $f(30n+25)$ and prove that $f(250n+125)\equiv 0\ (\text{mod\ 25}).$