In 1960 Rényi, in his Michigan State University lectures, asked for the number of random queries necessary to recover a hidden bijective labelling of n distinct objects. In each query one selects a random subset of labels and asks, which objects have these labels? We consider here an asymmetric version of the problem in which in every query an object is chosen with probability p > 1/2 and we ignore ‘inconclusive’ queries. We study the number of queries needed to recover the labelling in its entirety (Hn), before at least one element is recovered (Fn), and to recover a randomly chosen element (Dn). This problem exhibits several remarkable behaviours: Dn converges in probability but not almost surely; Hn and Fn exhibit phase transitions with respect to p in the second term. We prove that for p > 1/2 with high probability we need

$$H_n=\log_{1/p} n +{\tfrac{1}{2}} \log_{p/(1-p)}\log n +o(\log \log n)$$

$$ H_n=\log_{2} n +\sqrt{2 \log_{2} n} +o(\sqrt{\log n})$$

Let k ⩾ 2 be an integer. We show that if s = 2 and t ⩾ 2, or s = t = 3, then the maximum possible number of edges in a C2k+1-free graph containing no induced copy of Ks,t is asymptotically equal to (t − s + 1)1/s(n/2)2−1/s except when k = s = t = 2.

This strengthens a result of Allen, Keevash, Sudakov and Verstraëte [1], and answers a question of Loh, Tait, Timmons and Zhou [14].

Denote by ${\mathcal H}_k$(n, p) the random k-graph in which each k-subset of {1,. . .,n} is present with probability p, independent of other choices. More or less answering a question of Balogh, Bohman and Mubayi, we show: there is a fixed ε > 0 such that if n = 2k + 1 and p > 1 - ε, then w.h.p. (that is, with probability tending to 1 as k → ∞), ${\mathcal H}_k$(n, p) has the ‘Erdős–Ko–Rado property’. We also mention a similar random version of Sperner's theorem.

We study a restricted form of list colouring, for which every pair of lists that correspond to adjacent vertices may not share more than one colour. The optimal list size such that a proper list colouring is always possible given this restriction, we call separation choosability. We show for bipartite graphs that separation choosability increases with (the logarithm of) the minimum degree. This strengthens results of Molloy and Thron and, partially, of Alon. One attempt to drop the bipartiteness assumption precipitates a natural class of Ramsey-type questions, of independent interest. For example, does every triangle-free graph of minimum degree d contain a bipartite induced subgraph of minimum degree Ω(log d) as d→∞?

Let m(k) denote the maximum number of edges in a non-extendable, intersecting k-graph. Erdős and Lovász proved that m(k) ≤ kk. For k ≥ 625 we prove m(k) < kk・e−k1/4/6.

Mader proved that every strongly k-connected n-vertex digraph contains a strongly k-connected spanning subgraph with at most 2kn - 2k2 edges, where equality holds for the complete bipartite digraph DKk,n-k. For dense strongly k-connected digraphs, this upper bound can be significantly improved. More precisely, we prove that every strongly k-connected n-vertex digraph D contains a strongly k-connected spanning subgraph with at most kn + 800k(k + Δ(D)) edges, where Δ(D) denotes the maximum degree of the complement of the underlying undirected graph of a digraph D. Here, the additional term 800k(k + Δ(D)) is tight up to multiplicative and additive constants. As a corollary, this implies that every strongly k-connected n-vertex semicomplete digraph contains a strongly k-connected spanning subgraph with at most kn + 800k2 edges, which is essentially optimal since 800k2 cannot be reduced to the number less than k(k - 1)/2.

We also prove an analogous result for strongly k-arc-connected directed multigraphs. Both proofs yield polynomial-time algorithms.

For given graphs G1,…, Gk, the size-Ramsey number $\hat R({G_1}, \ldots ,{G_k})$ is the smallest integer m for which there exists a graph H on m edges such that in every k-edge colouring of H with colours 1,…,k, H contains a monochromatic copy of Gi of colour i for some 1 ≤ i ≤ k. We denote $\hat R({G_1}, \ldots ,{G_k})$ by ${\hat R_k}(G)$ when G1 = ⋯ = Gk = G.

Haxell, Kohayakawa and Łuczak showed that the size-Ramsey number of a cycle Cn is linear in n, ${\hat R_k}({C_n}) \le {c_k}n$ for some constant ck. Their proof, however, is based on Szemerédi’s regularity lemma so no specific constant ck is known.

In this paper, we give various upper bounds for the size-Ramsey numbers of cycles. We provide an alternative proof of ${\hat R_k}({C_n}) \le {c_k}n$ , avoiding use of the regularity lemma, where ck is exponential and doubly exponential in k, when n is even and odd, respectively. In particular, we show that for sufficiently large n we have ${\hat R_2}({C_n}) \le {10^5} \times cn$ , where c = 6.5 if n is even and c = 1989 otherwise.

We give combinatorial descriptions of two stochastic growth models for series-parallel networks introduced by Hosam Mahmoud by encoding the growth process via recursive tree structures. Using decompositions of the tree structures and applying analytic combinatorics methods allows a study of quantities in the corresponding series-parallel networks. For both models we obtain limiting distribution results for the degree of the poles and the length of a random source-to-sink path, and furthermore we get asymptotic results for the expected number of source-to-sink paths. Moreover, we introduce generalizations of these stochastic models by encoding the growth process of the networks via further important increasing tree structures.

We consider linear preferential attachment trees, and show that they can be regarded as random split trees in the sense of Devroye (1999), although with infinite potential branching. In particular, this applies to the random recursive tree and the standard preferential attachment tree. An application is given to the sum over all pairs of nodes of the common number of ancestors.

We upper-bound the number of common zeros over a finite grid of multivariate polynomials and an arbitrary finite collection of their consecutive Hasse derivatives (in a coordinate-wise sense). To that end, we make use of the tool from Gröbner basis theory known as footprint. Then we establish and prove extensions in this context of a family of well-known results in algebra and combinatorics. These include Alon's combinatorial Nullstellensatz [1], existence and uniqueness of Hermite interpolating polynomials over a grid, estimations of the parameters of evaluation codes with consecutive derivatives [20], and bounds on the number of zeros of a polynomial by DeMillo and Lipton [8], Schwartz [25], Zippel [26, 27] and Alon and Füredi [2]. As an alternative, we also extend the Schwartz-Zippel bound to weighted multiplicities and discuss its connection to our extension of the footprint bound.

As a strengthening of Hadwiger’s conjecture, Gerards and Seymour conjectured that every graph with no odd Kt minor is (t − 1)-colourable. We prove two weaker variants of this conjecture. Firstly, we show that for each t ⩾ 2, every graph with no odd Kt minor has a partition of its vertex set into 6t − 9 sets V1, …, V6t−9 such that each Vi induces a subgraph of bounded maximum degree. Secondly, we prove that for each t ⩾ 2, every graph with no odd Kt minor has a partition of its vertex set into 10t −13 sets V1,…, V10t−13 such that each Vi induces a subgraph with components of bounded size. The second theorem improves a result of Kawarabayashi (2008), which states that the vertex set can be partitioned into 496t such sets.

A family of sets is intersecting if no two of its members are disjoint, and has the Erdős–Ko–Rado property (or is EKR) if each of its largest intersecting subfamilies has non-empty intersection.

Denote by ${{\cal H}_k}(n,p)$ the random family in which each k-subset of {1, …, n} is present with probability p, independent of other choices. A question first studied by Balogh, Bohman and Mubayi asks:

\begin{equation} {\rm{For what }}p = p(n,k){\rm{is}}{{\cal H}_k}(n,p){\rm{likely to be EKR}}? \end{equation}

Here, for fixed c < 1/4, and $k \lt \sqrt {cn\log n} $ we give a precise answer to this question, characterizing those sequences p = p(n, k) for which

\begin{equation} {\mathbb{P}}({{\cal H}_k}(n,p){\rm{is EKR}}{\kern 1pt} ) \to 1{\rm{as }}n \to \infty . \end{equation}

We prove the following 30 year-old conjecture of Győri and Tuza: the edges of every n-vertex graph G can be decomposed into complete graphs C1,. . .,Cℓ of orders two and three such that |C1|+···+|Cℓ| ≤ (1/2+o(1))n2. This result implies the asymptotic version of the old result of Erdős, Goodman and Pósa that asserts the existence of such a decomposition with ℓ ≤ n2/4.

A set of points in d-dimensional Euclidean space is almost equidistant if, among any three points of the set, some two are at distance 1. We show that an almost-equidistant set in ℝd has cardinality O(d4/3).

Thirty years ago, the Robin Hood collision resolution strategy was introduced for open addressing hash tables, and a recurrence equation was found for the distribution of its search cost. Although this recurrence could not be solved analytically, it allowed for numerical computations that, remarkably, suggested that the variance of the search cost approached a value of 1.883 when the table was full. Furthermore, by using a non-standard mean-centred search algorithm, this would imply that searches could be performed in expected constant time even in a full table.

In spite of the time elapsed since these observations were made, no progress has been made in proving them. In this paper we introduce a technique to work around the intractability of the recurrence equation by solving instead an associated differential equation. While this does not provide an exact solution, it is sufficiently powerful to prove a bound of π2/3 for the variance, and thus obtain a proof that the variance of Robin Hood is bounded by a small constant for load factors arbitrarily close to 1. As a corollary, this proves that the mean-centred search algorithm runs in expected constant time.

We also use this technique to study the performance of Robin Hood hash tables under a long sequence of insertions and deletions, where deletions are implemented by marking elements as deleted. We prove that, in this case, the variance is bounded by 1/(1−α), where α is the load factor.

To model the behaviour of these hash tables, we use a unified approach that we apply also to study the First-Come-First-Served and Last-Come-First-Served collision resolution disciplines, both with and without deletions.

Small ball inequalities have been extensively studied in the setting of Gaussian processes and associated Banach or Hilbert spaces. In this paper, we focus on studying small ball probabilities for sums or differences of independent, identically distributed random elements taking values in very general sets. Depending on the setting – abelian or non-abelian groups, or vector spaces, or Banach spaces – we provide a collection of inequalities relating different small ball probabilities that are sharp in many cases of interest. We prove these distribution-free probabilistic inequalities by showing that underlying them are inequalities of extremal combinatorial nature, related among other things to classical packing problems such as the kissing number problem. Applications are given to moment inequalities.

We consider a problem introduced by Mossel and Ross (‘Shotgun assembly of labeled graphs’, arXiv:1504.07682). Suppose a random n × n jigsaw puzzle is constructed by independently and uniformly choosing the shape of each ‘jig’ from q possibilities. We are given the shuffled pieces. Then, depending on q, what is the probability that we can reassemble the puzzle uniquely? We say that two solutions of a puzzle are similar if they only differ by a global rotation of the puzzle, permutation of duplicate pieces, and rotation of rotationally symmetric pieces. In this paper, we show that, with high probability, such a puzzle has at least two non-similar solutions when 2 ⩽ q ⩽ 2e−1/2n, all solutions are similar when q ⩾ (2+ϵ)n, and the solution is unique when q = ω(n).

Let C be a bounded convex object in ℝd, and let P be a set of n points lying outside C. Further, let cp, cq be two integers with 1 ⩽ cq ⩽ cp ⩽ n - ⌊d/2⌋, such that every cp + ⌊d/2⌋ points of P contain a subset of size cq + ⌊d/2⌋ whose convex hull is disjoint from C. Then our main theorem states the existence of a partition of P into a small number of subsets, each of whose convex hulls are disjoint from C. Our proof is constructive and implies that such a partition can be computed in polynomial time.

In particular, our general theorem implies polynomial bounds for Hadwiger--Debrunner (p, q) numbers for balls in ℝd. For example, it follows from our theorem that when p > q = (1+β)⋅d/2 for β > 0, then any set of balls satisfying the (p, q)-property can be hit by O((1+β)2d2p1+1/β logp) points. This is the first improvement over a nearly 60 year-old exponential bound of roughly O(2d).

Our results also complement the results obtained in a recent work of Keller, Smorodinsky and Tardos where, apart from improvements to the bound on HD(p, q) for convex sets in ℝd for various ranges of p and q, a polynomial bound is obtained for regions with low union complexity in the plane.

Let k ⩾ 3 be a fixed integer. We exactly determine the asymptotic distribution of ln Zk(G(n, m)), where Zk(G(n, m)) is the number of k-colourings of the random graph G(n, m). A crucial observation to this end is that the fluctuations in the number of colourings can be attributed to the fluctuations in the number of small cycles in G(n, m). Our result holds for a wide range of average degrees, and for k exceeding a certain constant k0 it covers all average degrees up to the so-called condensation phase transition.

A tree functional is called additive if it satisfies a recursion of the form $F(T) = \sum_{j=1}^k F(B_j) + f(T)$, where B1, …, Bk are the branches of the tree T and f (T) is a toll function. We prove a general central limit theorem for additive functionals of d-ary increasing trees under suitable assumptions on the toll function. The same method also applies to generalized plane-oriented increasing trees (GPORTs). One of our main applications is a log-normal law that we prove for the size of the automorphism group of d-ary increasing trees, but other examples (old and new) are covered as well.