Given a set $L$ of $n$ lines in ${\mathbb R}^3$, joints are points in ${\mathbb R}^3$ that are incident to at least three non-coplanar lines in $L$. We show that there are at most $O(n^{5/3})$ incidences between $L$ and the set of its joints.

This result leads to related questions about incidences between $L$ and a set $P$ of $m$ points in ${\mathbb R}^3$. First, we associate with every point $p \in P$ the minimum number of planes it takes to cover all lines incident to $p$. Then the sum of these numbers is at most \[ O\big(m^{4/7}n^{5/7}+m+n\big).\] Second, if each line forms a fixed given non-zero angle with the $xy$-plane – we say the lines are equally inclined – then the number of (real) incidences is at most \[ O\big(\min\big\{m^{3/4}n^{1/2}\kappa(m),\ m^{4/7}n^{5/7}\big\} + m + n\big) , \] where $\kappa(m) \,{=}\, (\log m)^{O(\alpha^2(m))}$, and $\alpha(m)$ is the slowly growing inverse Ackermann function. These bounds are smaller than the tight Szemerédi–Trotter bound for point–line incidences in $\reals^2$, unless both bounds are linear. They are the first results of this type on incidences between points and $1$-dimensional objects in $\reals^3$. This research was stimulated by a question raised by G. Elekes.

This paper addresses the following question for a given graph H: What is the minimum number f(H) such that every graph with average degree at least f(H) contains H as a minor? Due to connections with Hadwiger's conjecture, this question has been studied in depth when H is a complete graph. Kostochka and Thomason independently proved that $f(K_t)=ct\sqrt{\ln t}$. More generally, Myers and Thomason determined f(H) when H has a super-linear number of edges. We focus on the case when H has a linear number of edges. Our main result, which complements the result of Myers and Thomason, states that if H has t vertices and average degree d at least some absolute constant, then $f(H)\leq 3.895\sqrt{\ln d}\,t$. Furthermore, motivated by the case when H has small average degree, we prove that if H has t vertices and q edges, then f(H) ⩽ t + 6.291q (where the coefficient of 1 in the t term is best possible).

A central result in extremal set theory is the celebrated theorem of Sperner from 1928, which gives the size of the largest family of subsets of [n] not containing a 2-chain, F1 ⊂ F2. Erdős extended this theorem to determine the largest family without a k-chain, F1 ⊂ F2 ⊂ . . . ⊂ Fk. Erdős and Katona, followed by Kleitman, asked how many chains must appear in families with sizes larger than the corresponding extremal bounds.

In 1966, Kleitman resolved this question for 2-chains, showing that the number of such chains is minimized by taking sets as close to the middle level as possible. Moreover, he conjectured the extremal families were the same for k-chains, for all k. In this paper, making the first progress on this problem, we verify Kleitman's conjecture for the families whose size is at most the size of the k + 1 middle levels. We also characterize all extremal configurations.

Suppose that X1, X2, . . . are independent identically distributed Bernoulli random variables with mean p. A Bernoulli factory for a function f takes as input X1, X2, . . . and outputs a random variable that is Bernoulli with mean f(p). A fast algorithm is a function that only depends on the values of X1, . . ., XT, where T is a stopping time with small mean. When f(p) is a real analytic function the problem reduces to being able to draw from linear functions Cp for a constant C > 1. Also it is necessary that Cp ⩽ 1 − ε for known ε > 0. Previous methods for this problem required extensive modification of the algorithm for every value of C and ε. These methods did not have explicit bounds on $\mathbb{E}[T]$ as a function of C and ε. This paper presents the first Bernoulli factory for f(p) = Cp with bounds on $\mathbb{E}[T]$ as a function of the input parameters. In fact, supp∈[0,(1−ε)/C]$\mathbb{E}[T]$ ≤ 9.5ε−1C. In addition, this method is very simple to implement. Furthermore, a lower bound on the average running time of any Cp Bernoulli factory is shown. For ε ⩽ 1/2, supp∈[0,(1−ε)/C]$\mathbb{E}[T]$≥0.004Cε−1, so the new method is optimal up to a constant in the running time.

A detachment of a hypergraph is formed by splitting each vertex into one or more subvertices, and sharing the incident edges arbitrarily among the subvertices. For a given edge-coloured hypergraph , we prove that there exists a detachment such that the degree of each vertex and the multiplicity of each edge in (and each colour class of ) are shared fairly among the subvertices in (and each colour class of , respectively).

Let be a hypergraph with vertex partition {V1,. . .,Vn}, |Vi| = pi for 1 ≤ i ≤ n such that there are λi edges of size hi incident with every hi vertices, at most one vertex from each part for 1 ≤ i ≤ m (so no edge is incident with more than one vertex of a part). We use our detachment theorem to show that the obvious necessary conditions for to be expressed as the union 1 ∪ ··· ∪ k of k edge-disjoint factors, where for 1 ≤ i ≤ k, i is ri-regular, are also sufficient. Baranyai solved the case of h1 = ··· = hm, λ1 = ··· = λm = 1, p1 = ··· = pm, r1 = ··· = rk. Berge and Johnson (and later Brouwer and Tijdeman, respectively) considered (and solved, respectively) the case of hi = i, 1 ≤ i ≤ m, p1 = ··· = pm = λ1 = ··· = λm = r1 = ··· = rk = 1. We also extend our result to the case where each i is almost regular.

We study the lower tail large deviation problem for subgraph counts in a random graph. Let XH denote the number of copies of H in an Erdős–Rényi random graph $\mathcal{G}(n,p)$. We are interested in estimating the lower tail probability $\mathbb{P}(X_H \le (1-\delta) \mathbb{E} X_H)$ for fixed 0 < δ < 1.

Thanks to the results of Chatterjee, Dembo and Varadhan, this large deviation problem has been reduced to a natural variational problem over graphons, at least for p ≥ n−αH (and conjecturally for a larger range of p). We study this variational problem and provide a partial characterization of the so-called ‘replica symmetric’ phase. Informally, our main result says that for every H, and 0 < δ < δH for some δH > 0, as p → 0 slowly, the main contribution to the lower tail probability comes from Erdős–Rényi random graphs with a uniformly tilted edge density. On the other hand, this is false for non-bipartite H and δ close to 1.

We prove that almost every r-regular digraph is Hamiltonian for all fixed r ≥ 3.

We consider a simple randomised algorithm that seeks a weak 2-colouring of a hypergraph H; that is, it tries to 2-colour the points of H so that no edge is monochromatic. If H has a particular well-behaved form of such a colouring, then the method is successful within expected number of iterations O(n3) when H has n points. In particular, when applied to a graph G with n nodes and chromatic number 3, the method yields a 2-colouring of the vertices such that no triangle is monochromatic in expected time O(n4).

Often when analysing randomized algorithms, especially parallel or distributed algorithms, one is called upon to show that some function of many independent choices is tightly concentrated about its expected value. For example, the algorithm might colour the vertices of a given graph with two colours and one would wish to show that, with high probability, very nearly half of all edges are monochromatic.

The classic result of Chernoff [3] gives such a large deviation result when the function is a sum of independent indicator random variables. The results of Hoeffding [5] and Azuma [2] give similar results for functions which can be expressed as martingales with a bounded difference property. Roughly speaking, this means that each individual choice has a bounded effect on the value of the function. McDiarmid [9] nicely summarized these results and gave a host of applications. Expressed a little differently, his main result is as follows.

We establish a generalization of the Expander Mixing Lemma for arbitrary (finite) simplicial complexes. The original lemma states that concentration of the Laplace spectrum of a graph implies combinatorial expansion (which is also referred to as mixing, or pseudo-randomness). Recently, an analogue of this lemma was proved for simplicial complexes of arbitrary dimension, provided that the skeleton of the complex is complete. More precisely, it was shown that a concentrated spectrum of the simplicial Hodge Laplacian implies a similar type of pseudo-randomness as in graphs. In this paper we remove the assumption of a complete skeleton, showing that simultaneous concentration of the Laplace spectra in all dimensions implies pseudo-randomness in any complex. We discuss various applications and present some open questions.

Let Δ ≥ 2 be a fixed integer. We show that the random graph ${\mathcal{G}_{n,p}}$ with $p\gg (\log n/n)^{1/\Delta}$ is robust with respect to the containment of almost spanning bipartite graphs H with maximum degree Δ and sublinear bandwidth in the following sense: asymptotically almost surely, if an adversary deletes arbitrary edges from ${\mathcal{G}_{n,p}}$ in such a way that each vertex loses less than half of its neighbours, then the resulting graph still contains a copy of all such H.

We show that any n-vertex complete graph with edges coloured with three colours contains a set of at most four vertices such that the number of the neighbours of these vertices in one of the colours is at least 2n/3. The previous best value, proved by Erdős, Faudree, Gould, Gyárfás, Rousseau and Schelp in 1989, is 22. It is conjectured that three vertices suffice.

As a consequence of an early result of Pach we show that every maximal triangle-free graph is either homomorphic with a member of a specific infinite sequence of graphs or contains the Petersen graph minus one vertex as a subgraph. From this result and further structural observations we derive that, if a (not necessarily maximal) triangle-free graph of order n has minimum degree δ[ges ]n/3, then the graph is either homomorphic with a member of the indicated family or contains the Petersen graph with one edge contracted. As a corollary we get a recent result due to Chen, Jin and Koh. Finally, we show that every triangle-free graph with δ>n/3 is either homomorphic with C5 or contains the Möbius ladder. A major tool is the observation that every triangle-free graph with δ[ges ]n/3 has a unique maximal triangle-free supergraph.

Let r = r(n) → ∞ with 3 [les ] r [les ] n1−η for an arbitrarily small constant η > 0, and let Gr denote a graph chosen uniformly at random from the set of r-regular graphs with vertex set {1, 2, …, n}. We prove that, with probability tending to 1 as n → ∞, Gr has the following properties: the independence number of Gr is asymptotically 2n log r/r and the chromatic number of Gr is asymptotically r/2nlogr.

Let 0 < p < 1, q = 1 − p and b be fixed and let G ∈ [Gscr ](n, p) be a graph on n vertices where each pair of vertices is joined independently with probability p. We show that the probability that every vertex of G has degree at most pn + b √npq is equal to (c + o(1))n, for c = c(b) the root of a certain equation. Surprisingly, c(0) = 0.6102 … is greater than ½ and c(b) is independent of p. To obtain these results we consider the complete graph on n vertices with weights on the edges. Taking these weights as independent normal N(p, pq) random variables gives a ‘continuous’ approximation to [Gscr ](n, p) whose degrees are much easier to analyse.

We study the number of subtrees on the fringe of random recursive trees and random binary search trees whose limit law is known to be either normal or Poisson or degenerate depending on the size of the subtree. We introduce a new approach to this problem which helps us to further clarify this phenomenon. More precisely, we derive optimal Berry–Esseen bounds and local limit theorems for the normal range and prove a Poisson approximation result as the subtree size tends to infinity.

We show that a set A ⊂ {0, 1}n with edge-boundary of size at most can be made into a subcube by at most (2ε/log2(1/ε))|A| additions and deletions, provided ε is less than an absolute constant.

We deduce that if A ⊂ {0, 1}n has size 2t for some t ∈ ℕ, and cannot be made into a subcube by fewer than δ|A| additions and deletions, then its edge-boundary has size at least provided δ is less than an absolute constant. This is sharp whenever δ = 1/2j for some j ∈ {1, 2, . . ., t}.

For 1 < α < 2 we derive the asymptotic distribution of the total length of external branches of a Beta(2 − α, α)-coalescent as the number n of leaves becomes large. It turns out that the fluctuations of the external branch length follow those of τn2−α over the entire parameter regime, where τn denotes the random number of coalescences that bring the n lineages down to one. This is in contrast to the fluctuation behaviour of the total branch length, which exhibits a transition at $\alpha_0 = (1+\sqrt 5)/2$ ([18]).

We provide a characterization of upper locally distributive lattices (ULD-lattices) in terms of edge colourings of their cover graphs. In many instances where a set of combinatorial objects carries the order structure of a lattice, this characterization yields a slick proof of distributivity or UL-distributivity. This is exemplified by proving a distributive lattice structure on Δ-bonds with invariant circular flow-difference. This instance generalizes several previously studied lattice structures, in particular, c-orientations (Propp), α-orientations of planar graphs (Felsner, resp. de Mendez) and planar flows (Khuller, Naor and Klein). The characterization also applies to other instances, e.g., to chip-firing games.

We consider Glauber dynamics on finite spin systems. The mixing time of Glauber dynamics can be bounded in terms of the influences of sites on each other. We consider three parameters bounding these influences: α, the total influence on a site, as studied by Dobrushin; α′, the total influence of a site, as studied by Dobrushin and Shlosman; and α″, the total influence of a site in any given context, which is related to the path-coupling method of Bubley and Dyer. It is known that if any of these parameters is less than 1 then random-update Glauber dynamics (in which a randomly chosen site is updated at each step) is rapidly mixing. It is also known that the Dobrushin condition α < 1 implies that systematic-scan Glauber dynamics (in which sites are updated in a deterministic order) is rapidly mixing. This paper studies two related issues, primarily in the context of systematic scan: (1) the relationship between the parameters α, α′ and α″, and (2) the relationship between proofs of rapid mixing using Dobrushin uniqueness (which typically use analysis techniques) and proofs of rapid mixing using path coupling. We use matrix balancing to show that the Dobrushin–Shlosman condition α′ < 1 implies rapid mixing of systematic scan. An interesting question is whether the rapid mixing results for scan can be extended to the α = 1 or α′ = 1 case. We give positive results for the rapid mixing of systematic scan for certain α = 1 cases. As an application, we show rapid mixing of systematic scan (for any scan order) for heat-bath Glauber dynamics for proper q-colourings of a degree-Δ graph G when q ≥ 2Δ.