We introduce the open degree of a compact space, and we show that for every natural number $n$, the separable Rosenthal compact spaces of degree $n$ have a finite basis.

The aim of this paper is to develop analytic techniques to deal with the monotonicity of certain combinatorial sequences. On the one hand, a criterion for the monotonicity of the function is given, which is a continuous analogue of a result of Wang and Zhu. On the other hand, the log-behaviour of the functions

is considered, where ζ(x) and Γ(x) are the Riemann zeta function and the Euler Gamma function, respectively. Consequently, the strict log-concavities of the function θ(x) (a conjecture of Chen et al.) and for some combinatorial sequences (including the Bernoulli numbers, the tangent numbers, the Catalan numbers, the Fuss–Catalan numbers, and the binomial coefficients are demonstrated. In particular, this contains some results of Chen et al., and Luca and Stănică. Finally, by researching the logarithmically complete monotonicity of some functions, the infinite log-monotonicity of the sequence

is proved. This generalizes two results of Chen et al. that both the Catalan numbers and the central binomial coefficients are infinitely log-monotonic, and strengthens one result of Su and Wang that is log-convex in n for positive integers d > δ. In addition, the asymptotically infinite log-monotonicity of derangement numbers is showed. In order to research the stronger properties of the above functions θ(x) and F(x), the logarithmically complete monotonicity of functions

is also obtained, which generalizes the results of Lee and Tepedelenlioǧlu, and Qi and Li.

We prove that proper coloring distinguishes between block factors and finitely dependent stationary processes. A stochastic process is finitely dependent if variables at sufficiently well-separated locations are independent; it is a block factor if it can be expressed as an equivariant finite-range function of independent variables. The problem of finding non-block-factor finitely dependent processes dates back to 1965. The first published example appeared in 1993, and we provide arguably the first natural examples. Schramm proved in 2008 that no stationary 1-dependent 3-coloring of the integers exists, and asked whether a $k$ -dependent $q$ -coloring exists for any $k$ and $q$ . We give a complete answer by constructing a 1-dependent 4-coloring and a 2-dependent 3-coloring. Our construction is canonical and natural, yet very different from all previous schemes. In its pure form it yields precisely the two finitely dependent colorings mentioned above, and no others. The processes provide unexpected connections between extremal cases of the Lovász local lemma and descent and peak sets of random permutations. Neither coloring can be expressed as a block factor, nor as a function of a finite-state Markov chain; indeed, no stationary finitely dependent coloring can be so expressed. We deduce extensions involving $d$ dimensions and shifts of finite type; in fact, any nondegenerate shift of finite type also distinguishes between block factors and finitely dependent processes.

Motivated by the local theory of Banach spaces, we introduce a notion of finite representability for metric spaces. This allows us to develop a new technique for comparing the generalised roundness of metric spaces. We illustrate this technique by applying it to Banach spaces and metric trees. In the realm of Banach spaces we obtain results such as the following: (1) if ${\mathcal{U}}$ is any ultrafilter and $X$ is any Banach space, then the second dual $X^{\ast \ast }$ and the ultrapower $(X)_{{\mathcal{U}}}$ have the same generalised roundness as $X$ , and (2) no Banach space of positive generalised roundness is uniformly homeomorphic to $c_{0}$ or $\ell _{p}$ , $2<p<\infty$ . For metric trees, we give the first examples of metric trees of generalised roundness one that have finite diameter. In addition, we show that metric trees of generalised roundness one possess special Euclidean embedding properties that distinguish them from all other metric trees.

We determine a bound for the valency in a family of dihedrants of twice odd prime orders which guarantees that the Cayley graphs are Ramanujan graphs. We take two families of Cayley graphs with the underlying dihedral group of order $2p$ : one is the family of all Cayley graphs and the other is the family of normal ones. In the normal case, which is easier, we discuss the problem for a wider class of groups, the Frobenius groups. The result for the family of all Cayley graphs is similar to that for circulants: the prime $p$ is ‘exceptional’ if and only if it is represented by one of six specific quadratic polynomials.

In this paper, we study the integer sequence $(E_{n})_{n\geq 1}$ , where $E_{n}$ counts the number of possible dimensions for centralisers of $n\times n$ matrices. We give an example to show another combinatorial interpretation of $E_{n}$ and present an implicit recurrence formula for $E_{n}$ , which may provide a fast algorithm for computing $E_{n}$ . Based on the recurrence, we obtain the asymptotic formula $E_{n}=\frac{1}{2}n^{2}-\frac{2}{3}\sqrt{2}n^{3/2}+O(n^{5/4})$ .

The classical theorem of Vizing states that every graph of maximum degree $d$ admits an edge coloring with at most $d+1$ colors. Furthermore, as it was earlier shown by Kőnig, $d$ colors suffice if the graph is bipartite. We investigate the existence of measurable edge colorings for graphings (or measure-preserving graphs). A graphing is an analytic generalization of a bounded-degree graph that appears in various areas, such as sparse graph limits, orbit equivalence and measurable group theory. We show that every graphing of maximum degree $d$ admits a measurable edge coloring with $d+O(\sqrt{d})$ colors; furthermore, if the graphing has no odd cycles, then $d+1$ colors suffice. In fact, if a certain conjecture about finite graphs that strengthens Vizing’s theorem is true, then our method will show that $d+1$ colors are always enough.

We say a graph is (Q n ,Q m )-saturated if it is a maximal Q m -free subgraph of the n-dimensional hypercube Q n . A graph is said to be (Q n ,Q m )-semi-saturated if it is a subgraph of Q n and adding any edge forms a new copy of Q m . The minimum number of edges a (Q n ,Q m )-saturated graph (respectively (Q n ,Q m )-semi-saturated graph) can have is denoted by sat(Q n ,Q m ) (respectively s-sat(Q n ,Q m )). We prove that

$$\begin{linenomath}\lim_{n\to\infty}\ffrac{\sat(Q_n,Q_m)}{e(Q_n)}=0,\end{linenomath}$$

$$\begin{linenomath}\ssat(Q_n,Q_m)\geq \ffrac{m+1}{2}\cdot 2^n,\end{linenomath}$$

Let G be an abelian group of cardinality n, where hcf(n, 6) = 1, and let A be a random subset of G. Form a graph ΓA on vertex set G by joining x to y if and only if x + y ∈ A. Then, with high probability as n → ∞, the chromatic number χ(ΓA ) is at most $(1 + o(1))\tfrac{n}{2\log_2 n}$ . This is asymptotically sharp when G = ℤ/nℤ, n prime.

For a given graph G of minimum degree at least k, let Gp denote the random spanning subgraph of G obtained by retaining each edge independently with probability p = p(k). We prove that if p ⩾ (logk + loglogk + ωk (1))/k, where ωk (1) is any function tending to infinity with k, then Gp asymptotically almost surely contains a cycle of length at least k + 1. When we take G to be the complete graph on k + 1 vertices, our theorem coincides with the classic result on the threshold probability for the existence of a Hamilton cycle in the binomial random graph.

Let r and d be positive integers with r<d. Consider a random d-ary tree constructed as follows. Start with a single vertex, and in each time-step choose a uniformly random leaf and give it d newly created offspring. Let 𝒯d,t be the tree produced after t steps. We show that there exists a fixed δ<1 depending on d and r such that almost surely for all large t, every r-ary subtree of 𝒯d,t has less than t δ vertices. The proof involves analysis that also yields a related result. Consider the following iterative construction of a random planar triangulation. Start with a triangle embedded in the plane. In each step, choose a bounded face uniformly at random, add a vertex inside that face and join it to the vertices of the face. In this way, one face is destroyed and three new faces are created. After t steps, we obtain a random triangulated plane graph with t+3 vertices, which is called a random Apollonian network. We prove that there exists a fixed δ<1, such that eventually every path in this graph has length less than t 𝛿, which verifies a conjecture of Cooper and Frieze (2015).

In this paper we study random Apollonian networks (RANs) and evolving Apollonian networks (EANs), in d dimensions for any d≥2, i.e. dynamically evolving random d-dimensional simplices, looked at as graphs inside an initial d-dimensional simplex. We determine the limiting degree distribution in RANs and show that it follows a power-law tail with exponent τ=(2d-1)/(d-1). We further show that the degree distribution in EANs converges to the same degree distribution if the simplex-occupation parameter in the nth step of the dynamics tends to 0 but is not summable in n. This result gives a rigorous proof for the conjecture of Zhang et al. (2006) that EANs tend to exhibit similar behaviour as RANs once the occupation parameter tends to 0. We also determine the asymptotic behaviour of the shortest paths in RANs and EANs for any d≥2. For RANs we show that the shortest path between two vertices chosen u.a.r. (typical distance), the flooding time of a vertex chosen uniformly at random, and the diameter of the graph after n steps all scale as a constant multiplied by log n. We determine the constants for all three cases and prove a central limit theorem for the typical distances. We prove a similar central limit theorem for typical distances in EANs.

Given any two vertices u, v of a random geometric graph G(n, r), denote by d E (u, v) their Euclidean distance and by d E (u, v) their graph distance. The problem of finding upper bounds on d G (u, v) conditional on d E (u, v) that hold asymptotically almost surely has received quite a bit of attention in the literature. In this paper we improve the known upper bounds for values of r=ω(√logn) (that is, for r above the connectivity threshold). Our result also improves the best known estimates on the diameter of random geometric graphs. We also provide a lower bound on d E (u, v) conditional on d E (u, v).

We give sufficient conditions for a graph to be traceable and Hamiltonian in terms of the Wiener index and the complement of the graph, which correct and extend the result of Yang [‘Wiener index and traceable graphs’, Bull. Aust. Math. Soc.88 (2013), 380–383]. We also present sufficient conditions for a bipartite graph to be traceable and Hamiltonian in terms of its Wiener index and quasicomplement. Finally, we give sufficient conditions for a graph or a bipartite graph to be traceable and Hamiltonian in terms of its distance spectral radius.

We prove that triangular configurations are plentiful in large subsets of Cartesian squares of finite quasirandom groups from classes having the quasirandom ultraproduct property, for example the class of finite simple groups. This is deduced from a strong double recurrence theorem for two commuting measure-preserving actions of a minimally almost periodic (not necessarily amenable or locally compact) group on a (not necessarily separable) probability space.

We study the lower tail large deviation problem for subgraph counts in a random graph. Let X H 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 −α H (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.

Judicious partitioning problems on graphs and hypergraphs ask for partitions that optimize several quantities simultaneously. Let k ≥ 2 be an integer and let G be a hypergraph with m i edges of size i for i=1,2. Bollobás and Scott conjectured that G has a partition into k classes, each of which contains at most $m_1/k+m_2/k^2+O(\sqrt{m_1+m_2})$ edges. In this paper, we confirm the conjecture affirmatively by showing that G has a partition into k classes, each of which contains at most

$$m_1/k+m_2/k^2+\ffrac{k-1}{2k^2}\sqrt{2(km_1+m_2)}+O(1)$$.

We consider the Widom–Rowlinson model of two types of interacting particles on d-regular graphs. We prove a tight upper bound on the occupancy fraction, the expected fraction of vertices occupied by a particle under a random configuration from the model. The upper bound is achieved uniquely by unions of complete graphs on d + 1 vertices, K d+1. As a corollary we find that K d+1 also maximizes the normalized partition function of the Widom–Rowlinson model over the class of d-regular graphs. A special case of this shows that the normalized number of homomorphisms from any d-regular graph G to the graph HWR , a path on three vertices with a loop on each vertex, is maximized by K d+1. This proves a conjecture of Galvin.

A Latin square is reduced if its first row and first column are in natural order. For Latin squares of a particular order $n$ , there are four possible different parities. We confirm a conjecture of Stones and Wanless by showing asymptotic equality between the numbers of reduced Latin squares of each possible parity as the order $n\rightarrow \infty$ .