We study approximations for the Lévy area of Brownian motion which are based on the Fourier series expansion and a polynomial expansion of the associated Brownian bridge. Comparing the asymptotic convergence rates of the Lévy area approximations, we see that the approximation resulting from the polynomial expansion of the Brownian bridge is more accurate than the Kloeden–Platen–Wright approximation, whilst still only using independent normal random vectors. We then link the asymptotic convergence rates of these approximations to the limiting fluctuations for the corresponding series expansions of the Brownian bridge. Moreover, and of interest in its own right, the analysis we use to identify the fluctuation processes for the Karhunen–Loève and Fourier series expansions of the Brownian bridge is extended to give a stand-alone derivation of the values of the Riemann zeta function at even positive integers.

We introduce a notion of the derivative with respect to a function, not necessarily related to a probability distribution, which generalizes the concept of derivative as proposed by Lebesgue [14]. The differential calculus required to solve linear differential equations using this notion of the derivative is included in the paper. The definition given here may also be considered as the inverse operator of a modified notion of the Riemann–Stieltjes integral. Both this unified approach and the results of differential calculus allow us to characterize distributions in terms of three different types of conditional expectations. In applying these results, a test of goodness of fit is also indicated. Finally, two characterizations of a general Poisson process are included. Specifically, a useful result for the homogeneous Poisson process is generalized.

We make the first steps towards generalising the theory of stochastic block models, in the sparse regime, towards a model where the discrete community structure is replaced by an underlying geometry. We consider a geometric random graph over a homogeneous metric space where the probability of two vertices to be connected is an arbitrary function of the distance. We give sufficient conditions under which the locations can be recovered (up to an isomorphism of the space) in the sparse regime. Moreover, we define a geometric counterpart of the model of flow of information on trees, due to Mossel and Peres, in which one considers a branching random walk on a sphere and the goal is to recover the location of the root based on the locations of leaves. We give some sufficient conditions for percolation and for non-percolation of information in this model.

For fixed positive integers $k,q,r$ with $q$ a prime power and large $m$, we investigate matrices with $m$ rows and a maximum number $N_q (m,k,r)$ of columns, such that each column contains at most $r$ nonzero entries from the finite field $GF(q)$ and any $k$ columns are linearly independent over $GF(q)$. For even integers $k \geq 2$ we obtain the lower bounds $N_q(m,k,r) = \Omega (m^{kr/(2(k-1))})$, and $N_q(m,k,r) = \Omega (m^{((k-1)r)/(2(k-2))})$ for odd $k \geq 3$. For $k=2^i$ we show that $N_q(m,k,r) = \Theta ( m^{kr/(2(k-1))})$ if $\gcd(k-1,r) = k-1$, while for arbitrary even $k \geq 4$ with $\gcd(k-1,r) =1$ we have $N_q(m,k,r) = \Omega (m^{kr/(2(k-1))} \cdot (\log m)^{1/(k-1)})$. Matrices which fulfil these lower bounds can be found in polynomial time. Moreover, for $\Char (GF(q)) > 2 $ we obtain $N_q(m,4,r) = \Theta (m^{\lceil 4r/3\rceil/2})$, while for $\Char (GF(q)) = 2$ we can only show that $N_q(m,4,r) = O (m^{\lceil 4r/3\rceil/2})$. Our results extend and complement earlier results from [7, 18], where the case $q=2$ was considered.

Let H be a graph, and let CH(G) be the number of (subgraph isomorphic) copies of H contained in a graph G. We investigate the fundamental problem of estimating CH(G). Previous results cover only a few specific instances of this general problem, for example the case when H has degree at most one (the monomer-dimer problem). In this paper we present the first general subcase of the subgraph isomorphism counting problem, which is almost always efficiently approximable. The results rely on a new graph decomposition technique. Informally, the decomposition is a labelling of the vertices such that every edge is between vertices with different labels, and for every vertex all neighbours with a higher label have identical labels. The labelling implicitly generates a sequence of bipartite graphs, which permits us to break the problem of counting embeddings of large subgraphs into that of counting embeddings of small subgraphs. Using this method, we present a simple randomized algorithm for the counting problem. For all decomposable graphs H and all graphs G, the algorithm is an unbiased estimator. Furthermore, for all graphs H having a decomposition where each of the bipartite graphs generated is small and almost all graphs G, the algorithm is a fully polynomial randomized approximation scheme.

We show that the graph classes of H for which we obtain a fully polynomial randomized approximation scheme for almost all G includes graphs of degree at most two, bounded-degree forests, bounded-width grid graphs, subdivision of bounded-degree graphs, and major subclasses of outerplanar graphs, series-parallel graphs and planar graphs of large girth, whereas unbounded-width grid graphs are excluded. Moreover, our general technique can easily be applied to proving many more similar results.

Suppose k is a positive integer and ${\cal X}$ is a k-fold packing of the plane by infinitely many arc-connected compact sets, which means that every point of the plane belongs to at most k sets. Suppose there is a function f(n) = o(n2) with the property that any n members of ${\cal X}$ determine at most f(n) holes, which means that the complement of their union has at most f(n) bounded connected components. We use tools from extremal graph theory and the topological Helly theorem to prove that ${\cal X}$ can be decomposed into at most p (1-fold) packings, where p is a constant depending only on k and f.

Suppose each vertex of a bipartite multigraph (with partition (X, Y)) is assigned a set of colours; we say this colour scheme is feasible if the edges of the graph can be properly coloured so that each receives a colour in both of its endpoints' sets. We prove various results showing that certain types of colour scheme are always feasible. For instance, we prove that if the colour scheme obtained by assigning the set {1,…, d(x)} of colours to each vertex x of X and the set T = {1,…, t} (t < Δ(X)) to each vertex of Y is feasible, then so is every colour scheme where each vertex x of X gets d(x) colours from T and each vertex of Y gets the set T.

An analytic study is made of the correlation structure of Tausworthe and linear congruential random number generators. The former case is analyzed by the bit mask correlations recently introduced by Compagner. The latter is studied first by an extension to word masks, which include spectral test coefficients as special cases, and then by the bit mask procedure. Although low order bit mask coefficients vanish in both cases, the Tausworthe generator appears to produce a substantially smaller non-vanishing correlation set for large masks – but with larger correlation values – than does the linear congruential.

Let D(G) be the smallest quantifier depth of a first-order formula which is true for a graph G but false for any other non-isomorphic graph. This can be viewed as a measure for the descriptive complexity of G in first-order logic.

We show that almost surely , where G is a random tree of order n or the giant component of a random graph with constant c<1. These results rely on computing the maximum of D(T) for a tree T of order n and maximum degree l, so we study this problem as well.

A 3-connected graph $G$ is weakly 3-connected if, for every edge $e$ of $G$, at most one of $G\backslash e$ and $G/e$ is 3-connected. The main result of this paper is that any weakly 3-connected graph can be reduced to $K_4$ by a sequence of simple operations. This extends a result of Dawes [5] on minimally 3-connected graphs.

The number of excluded minors for the class of graphs with path-width at most two is very large. To give a practical characterization of the obstructions, we introduce some operations which preserve path-width at most two. We give a list of ten graphs such that any graph with path-width more than two can be reduced – by taking minors and applying our operations – to one of the graphs on our list. We think that our operations and excluded substructures give a far more transparent description of the class of graphs with path-width at most two than Kinnersley and Langston's characterization by 110 excluded minors (see [4]).

We establish a fundamental property of bivariate Pareto records for independent observations uniformly distributed in the unit square. We prove that the asymptotic conditional distribution of the number of records broken by an observation given that the observation sets a record is Geometric with parameter 1/2.

We study sum-free sets in sparse random subsets of even-order abelian groups. In particular, we determine the sharp threshold for the following property: the largest such set is contained in some maximum-size sum-free subset of the group. This theorem extends recent work of Balogh, Morris and Samotij, who resolved the case G = ℤ2n, and who obtained a weaker threshold (up to a constant factor) in general.

A graph G arrows a graph H if in every 2-edge-colouring of G there exists a monochromatic copy of H. Schelp had the idea that if the complete graph $K_n$ arrows a small graph H, then every ‘dense’ subgraph of $K_n$ also arrows H, and he outlined some problems in this direction. Our main result is in this spirit. We prove that for every sufficiently large n, if $n = 3t+r$ where $r \in \{0,1,2\}$ and G is an n-vertex graph with $\delta(G) \ge (3n-1)/4$ , then for every 2-edge-colouring of G, either there are cycles of every length $\{3, 4, 5, \dots, 2t+r\}$ of the same colour, or there are cycles of every even length $\{4, 6, 8, \dots, 2t+2\}$ of the samecolour.

Our result is tight in the sense that no longer cycles (of length $>2t+r$ ) can be guaranteed and the minimum degree condition cannot be reduced. It also implies the conjecture of Schelp that for every sufficiently large n, every $(3t-1)$ -vertex graph G with minimum degree larger than $3|V(G)|/4$ arrows the path $P_{2n}$ with 2n vertices. Moreover, it implies for sufficiently large n the conjecture by Benevides, Łuczak, Scott, Skokan and White that for $n=3t+r$ where $r \in \{0,1,2\}$ and every n-vertex graph G with $\delta(G) \ge 3n/4$ , in each 2-edge-colouring of G there exists a monochromatic cycle of length at least $2t+r$ .

Let ${\bf C}$ denote the field of complex numbers and $\Omega_n$ the set of $n$th roots of unity. For $t = 0,\ldots,n-1$, define the ideal $\Im(n,t+1) \subset {\bf C}[x_0,\ldots,x_{t}]$ consisting of those polynomials in $t+1$ variables that vanish on distinct $n$th roots of unity; that is, $f \in \Im(n,t+1)$ if and only if $f(\omega_0,\ldots,\omega_{t}) = 0$ for all $(\omega_0,\ldots,\omega_{t}) \in \Omega_n^{t+1}$ satisfying $\omega_i \neq \omega_j$, for $0 \le i < j \le t$.

In this paper we apply Gröbner basis methods to give a Combinatorial Nullstellensatz characterization of the ideal $\Im(n,t+1)$. In particular, if $f \in {\bf C}[x_0,\ldots,x_{t}]$, then we give a necessary and sufficient condition on the coefficients of $f$ for membership in $\Im(n,t+1)$.

We prove that, with high probability, in every 2-edge-colouring of the random tournament on n vertices there is a monochromatic copy of every oriented tree of order $O(n{\rm{/}}\sqrt {{\rm{log}} \ n} )$. This generalizes a result of the first, third and fourth authors, who proved the same statement for paths, and is tight up to a constant factor.

We show that the scenery reconstruction problem on the Boolean hypercube is in general impossible. This is done by using locally biased functions, in which every vertex has a constant fraction of neighbours coloured by 1, and locally stable functions, in which every vertex has a constant fraction of neighbours coloured by its own colour. Our methods are constructive, and also give super-polynomial lower bounds on the number of locally biased and locally stable functions. We further show similar results for ℤn and other graphs, and offer several follow-up questions.

Given a graph $H$ and a positive integer $n$ , the Turán number $\mathrm{ex}(n,H)$ is the maximum number of edges in an $n$ -vertex graph that does not contain $H$ as a subgraph. A real number $r\in (1,2)$ is called a Turán exponent if there exists a bipartite graph $H$ such that $\mathrm{ex}(n,H)=\Theta (n^r)$ . A long-standing conjecture of Erdős and Simonovits states that $1+\frac{p}{q}$ is a Turán exponent for all positive integers $p$ and $q$ with $q\gt p$ .

In this paper, we show that $1+\frac{p}{q}$ is a Turán exponent for all positive integers $p$ and $q$ with $q \gt p^{2}$ . Our result also addresses a conjecture of Janzer [18].

We study two models of discrete height functions, that is, models of random integer-valued functions on the vertices of a tree. First, we consider the random homomorphism model, in which neighbours must have a height difference of exactly one. The local law is uniform by definition. We prove that the height variance of this model is bounded, uniformly over all boundary conditions (both in terms of location and boundary heights). This implies a strong notion of localisation, uniformly over all extremal Gibbs measures of the system. For the second model, we consider directed trees, in which each vertex has exactly one parent and at least two children. We consider the locally uniform law on height functions which are monotone, that is, such that the height of the parent vertex is always at least the height of the child vertex. We provide a complete classification of all extremal gradient Gibbs measures, and describe exactly the localisation-delocalisation transition for this model. Typical extremal gradient Gibbs measures are localised also in this case. Localisation in both models is consistent with the observation that the Gaussian free field is localised on trees, which is an immediate consequence of transience of the random walk.

The Turán number ex(n, H) of a graph H is the maximal number of edges in an H-free graph on n vertices. In 1983, Chung and Erdős asked which graphs H with e edges minimise ex(n, H). They resolved this question asymptotically for most of the range of e and asked to complete the picture. In this paper, we answer their question by resolving all remaining cases. Our result translates directly to the setting of universality, a well-studied notion of finding graphs which contain every graph belonging to a certain family. In this setting, we extend previous work done by Babai, Chung, Erdős, Graham and Spencer, and by Alon and Asodi.