If ${\mathfrak {F}}$ is a type-definable family of commensurable subsets, subgroups or subvector spaces in a metric structure, then there is an invariant subset, subgroup or subvector space commensurable with ${\mathfrak {F}}$. This in particular applies to type-definable or hyper-definable objects in a classical first-order structure.

We introduce a class of non-uniform random recursive trees grown with an attachment preference for young age. Via the Chen–Stein method of Poisson approximation, we find that the outdegree of a node is characterized in the limit by ‘perturbed’ Poisson laws, and the perturbation diminishes as the node index increases. As the perturbation is attenuated, a pure Poisson limit ultimately emerges in later phases. Moreover, we derive asymptotics for the proportion of leaves and show that the limiting fraction is less than one half. Finally, we study the insertion depth in a random tree in this class. For the insertion depth, we find the exact probability distribution, involving Stirling numbers, and consequently we find the exact and asymptotic mean and variance. Under appropriate normalization, we derive a concentration law and a limiting normal distribution. Some of these results contrast with their counterparts in the uniform attachment model, and some are similar.

This paper studies the scaling of the expected total queue size in an $n\times n$ input-queued switch, as a function of both the load $\rho$ and the system scale n. We provide a new class of scheduling policies under which the expected total queue size scales as $O\big( n(1-\rho)^{-4/3} \log \big(\!\max\big\{\frac{1}{1-\rho}, n\big\}\big)\big)$, over all n and $\rho<1$, when the arrival rates are uniform. This improves on the best previously known scalings in two regimes: $O\big(n^{1.5}(1-\rho)^{-1} \log \frac{1}{1-\rho}\big)$ when $\Omega\big(n^{-1.5}\big) \le 1-\rho \le O\big(n^{-1}\big)$ and $O\big(\frac{n\log n}{(1-\rho)^2}\big)$ when $1-\rho \geq \Omega(n^{-1})$. A key ingredient in our method is a tight characterization of the largest k-factor of a random bipartite multigraph, which may be of independent interest.

In this paper we study first passage percolation on a random graph model, the configuration model. We first introduce the notions of weighted diameter, which is the maximum of the weighted lengths of all optimal paths between any two vertices in the graph, and the flooding time, which represents the time (weighted length) needed to reach all the vertices in the graph starting from a uniformly chosen vertex. Our result consists in describing the asymptotic behavior of the diameter and the flooding time, as the number of vertices n tends to infinity, in the case where the weight distribution G has an exponential tail behavior, and proving that this category of distributions is the largest possible for which the asymptotic behavior holds.

We prove a ‘resilience’ version of Dirac’s theorem in the setting of random regular graphs. More precisely, we show that whenever d is sufficiently large compared to $\epsilon > 0$, a.a.s. the following holds. Let $G'$ be any subgraph of the random n-vertex d-regular graph $G_{n,d}$ with minimum degree at least $$(1/2 + \epsilon )d$$. Then $G'$ is Hamiltonian.

This proves a conjecture of Ben-Shimon, Krivelevich and Sudakov. Our result is best possible: firstly the condition that d is large cannot be omitted, and secondly the minimum degree bound cannot be improved.

Erdős, Gyárfás and Pyber showed that every r-edge-coloured complete graph Kn can be covered by 25 r2 log r vertex-disjoint monochromatic cycles (independent of n). Here we extend their result to the setting of binomial random graphs. That is, we show that if $p= p(n) = \Omega(n^{-1/(2r)})$, then with high probability any r-edge-coloured G(n, p) can be covered by at most 1000r4 log r vertex-disjoint monochromatic cycles. This answers a question of Korándi, Mousset, Nenadov, Škorić and Sudakov.

We prove that any n-vertex graph whose complement is triangle-free contains n2/12 – o(n2) edge-disjoint triangles. This is tight for the disjoint union of two cliques of order n/2. We also prove a corresponding stability theorem, that all large graphs attaining the above bound are close to being bipartite. Our results answer a question of Alon and Linial, and make progress on a conjecture of Erdős.

Several discrete geometry problems are equivalent to estimating the size of the largest homogeneous sets in graphs that happen to be the union of few comparability graphs. An important observation for such results is that if G is an n-vertex graph that is the union of r comparability (or more generally, perfect) graphs, then either G or its complement contains a clique of size $n^{1/(r+1)}$.

This bound is known to be tight for $r=1$. The question whether it is optimal for $r\ge 2$ was studied by Dumitrescu and Tóth. We prove that it is essentially best possible for $r=2$, as well: we introduce a probabilistic construction of two comparability graphs on n vertices, whose union contains no clique or independent set of size $n^{1/3+o(1)}$.

Using similar ideas, we can also construct a graph G that is the union of r comparability graphs, and neither G nor its complement contain a complete bipartite graph with parts of size $cn/{(log n)^r}$. With this, we improve a result of Fox and Pach.

In this work we consider three well-studied broadcast protocols: push, pull and push&pull. A key property of all these models, which is also an important reason for their popularity, is that they are presumed to be very robust, since they are simple, randomized and, crucially, do not utilize explicitly the global structure of the underlying graph. While sporadic results exist, there has been no systematic theoretical treatment quantifying the robustness of these models. Here we investigate this question with respect to two orthogonal aspects: (adversarial) modifications of the underlying graph and message transmission failures.

We explore in particular the following notion of local resilience: beginning with a graph, we investigate up to which fraction of the edges an adversary may delete at each vertex, so that the protocols need significantly more rounds to broadcast the information. Our main findings establish a separation among the three models. On one hand, pull is robust with respect to all parameters that we consider. On the other hand, push may slow down significantly, even if the adversary may modify the degrees of the vertices by an arbitrarily small positive fraction only. Finally, push&pull is robust when no message transmission failures are considered, otherwise it may be slowed down.

On the technical side, we develop two novel methods for the analysis of randomized rumour-spreading protocols. First, we exploit the notion of self-bounding functions to facilitate significantly the round-based analysis: we show that for any graph the variance of the growth of informed vertices is bounded by its expectation, so that concentration results follow immediately. Second, in order to control adversarial modifications of the graph we make use of a powerful tool from extremal graph theory, namely Szemerédi’s Regularity Lemma.

An ordered hypergraph is a hypergraph whose vertex set is linearly ordered, and a convex geometric hypergraph is a hypergraph whose vertex set is cyclically ordered. Extremal problems for ordered and convex geometric graphs have a rich history with applications to a variety of problems in combinatorial geometry. In this paper, we consider analogous extremal problems for uniform hypergraphs, and determine the order of magnitude of the extremal function for various ordered and convex geometric paths and matchings. Our results generalize earlier works of Braß–Károlyi–Valtr, Capoyleas–Pach, and Aronov–Dujmovič–Morin–Ooms-da Silveira. We also provide a new variation of the Erdős-Ko-Rado theorem in the ordered setting.

A well-known observation of Lovász is that if a hypergraph is not 2-colourable, then at least one pair of its edges intersect at a single vertex. In this short paper we consider the quantitative version of Lovász’s criterion. That is, we ask how many pairs of edges intersecting at a single vertex should belong to a non-2-colourable n-uniform hypergraph. Our main result is an exact answer to this question, which further characterizes all the extremal hypergraphs. The proof combines Bollobás’s two families theorem with Pluhar’s randomized colouring algorithm.

By taking square lattices as a two-dimensional analogue to Beatty sequences, we are motivated to define and explore the notion of complementary lattices. In particular, we present a continuous one-parameter family of complementary lattices. This main result then yields several novel examples of complementary sequences, along with a geometric proof of the fundamental property of Beatty sequences.

We give a fully polynomial-time randomized approximation scheme (FPRAS) for the number of bases in bicircular matroids. This is a natural class of matroids for which counting bases exactly is #P-hard and yet approximate counting can be done efficiently.

We consider maximal non-l-intertwining collections, which are a higher-dimensional version of the maximal non-crossing collections which give clusters of Plücker coordinates in the Grassmannian coordinate ring, as described by Scott. We extend a method of Scott for producing such collections, which are related to tensor products of higher Auslander algebras of type A. We show that a higher preprojective algebra of the tensor product of two d-representation-finite algebras has a d-precluster-tilting subcategory. Finally, we relate mutations of these collections to a form of tilting for these algebras.

For positive integers n and d > 0, let $G(\mathbb {Q}^n,\; d)$ denote the graph whose vertices are the set of rational points $\mathbb {Q}^n$, with $u,v \in \mathbb {Q}^n$ being adjacent if and only if the Euclidean distance between u and v is equal to d. Such a graph is deemed “non-trivial” if d is actually realized as a distance between points of $\mathbb {Q}^n$. In this paper, we show that a space $\mathbb {Q}^n$ has the property that all pairs of non-trivial distance graphs $G(\mathbb {Q}^n,\; d_1)$ and $G(\mathbb {Q}^n,\; d_2)$ are isomorphic if and only if n is equal to 1, 2, or a multiple of 4. Along the way, we make a number of observations concerning the clique number of $G(\mathbb {Q}^n,\; d)$.

Given a fixed graph H, a real number p(0, 1) and an infinite Erdös–Rényi graph G ∼ G(∞, p), how many adjacency queries do we have to make to find a copy of H inside G with probability at least 1/2? Determining this number f(H, p) is a variant of the subgraph query problem introduced by Ferber, Krivelevich, Sudakov and Vieira. For every graph H, we improve the trivial upper bound of f(H, p) = O(p−d), where d is the degeneracy of H, by exhibiting an algorithm that finds a copy of H in time O(p−d) as p goes to 0. Furthermore, we prove that there are 2-degenerate graphs which require p−2+o(1) queries, showing for the first time that there exist graphs H for which f(H, p) does not grow like a constant power of p−1 as p goes to 0. Finally, we answer a question of Feige, Gamarnik, Neeman, Rácz and Tetali by showing that for any δ < 2, there exists α < 2 such that one cannot find a clique of order α log2 n in G(n, 1/2) in nδ queries.

We show that the Mallows measure on permutations of $1,\dots ,n$ arises as the law of the unique Gale–Shapley stable matching of the random bipartite graph with vertex set conditioned to be perfect, where preferences arise from the natural total ordering of the vertices of each gender but are restricted to the (random) edges of the graph. We extend this correspondence to infinite intervals, for which the situation is more intricate. We prove that almost surely, every stable matching of the random bipartite graph obtained by performing Bernoulli percolation on the complete bipartite graph $K_{{\mathbb Z},{\mathbb Z}}$ falls into one of two classes: a countable family $(\sigma _n)_{n\in {\mathbb Z}}$ of tame stable matchings, in which the length of the longest edge crossing k is $O(\log |k|)$ as $k\to \pm \infty $, and an uncountable family of wild stable matchings, in which this length is $\exp \Omega (k)$ as $k\to +\infty $. The tame stable matching $\sigma _n$ has the law of the Mallows permutation of ${\mathbb Z}$ (as constructed by Gnedin and Olshanski) composed with the shift $k\mapsto k+n$. The permutation $\sigma _{n+1}$ dominates $\sigma _{n}$ pointwise, and the two permutations are related by a shift along a random strictly increasing sequence.

The triangle packing number v(G) of a graph G is the maximum size of a set of edge-disjoint triangles in G. Tuza conjectured that in any graph G there exists a set of at most 2v(G) edges intersecting every triangle in G. We show that Tuza’s conjecture holds in the random graph G = G(n, m), when m ⩽ 0.2403n3/2 or m ⩾ 2.1243n3/2. This is done by analysing a greedy algorithm for finding large triangle packings in random graphs.

A celebrated theorem of Pippenger states that any almost regular hypergraph with small codegrees has an almost perfect matching. We show that one can find such an almost perfect matching which is ‘pseudorandom’, meaning that, for instance, the matching contains as many edges from a given set of edges as predicted by a heuristic argument.

We show that, for a constant-degree algebraic curve γ in ℝD, every set of n points on γ spans at least Ω(n4/3) distinct distances, unless γ is an algebraic helix, in the sense of Charalambides [2]. This improves the earlier bound Ω(n5/4) of Charalambides [2].

We also show that, for every set P of n points that lie on a d-dimensional constant-degree algebraic variety V in ℝD, there exists a subset S ⊂ P of size at least Ω(n4/(9+12(d−1))), such that S spans $\left({\begin{array}{*{20}{c}} {|S|} \\ 2 \\\end{array}} \right)$ distinct distances. This improves the earlier bound of Ω(n1/(3d)) of Conlon, Fox, Gasarch, Harris, Ulrich and Zbarsky [4].

Both results are consequences of a common technical tool.