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 log2n in G(n, 1/2) in nδ queries.

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.

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.

We introduce a non-increasing tree growth process $((T_n,{\sigma}_n),\, n\ge 1)$ , where Tn is a rooted labelled tree on n vertices and σn is a permutation of the vertex labels. The construction of (Tn, σn) from (Tn−1, σn−1) involves rewiring a random (possibly empty) subset of edges in Tn−1 towards the newly added vertex; as a consequence Tn−1 ⊄ Tn with positive probability. The key feature of the process is that the shape of Tn has the same law as that of a random recursive tree, while the degree distribution of any given vertex is not monotone in the process.

We present two applications. First, while couplings between Kingman’s coalescent and random recursive trees were known for any fixed n, this new process provides a non-standard coupling of all finite Kingman’s coalescents. Second, we use the new process and the Chen–Stein method to extend the well-understood properties of degree distribution of random recursive trees to extremal-range cases. Namely, we obtain convergence rates on the number of vertices with degree at least $c\ln n$ , c ∈ (1, 2), in trees with n vertices. Further avenues of research are discussed.

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 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.

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.