Simple families of increasing trees were introduced by Bergeron, Flajolet and Salvy. They include random binary search trees, random recursive trees and random plane-oriented recursive trees (PORTs) as important special cases. In this paper, we investigate the number of subtrees of size k on the fringe of some classes of increasing trees, namely generalized PORTs and d-ary increasing trees. We use a complex-analytic method to derive precise expansions of mean value and variance as well as a central limit theorem for fixed k. Moreover, we propose an elementary approach to derive limit laws when k is growing with n. Our results have consequences for the occurrence of pattern sizes on the fringe of increasing trees.

Let H be a graph on n vertices and let the blow-up graph G[H] be defined as follows. We replace each vertex vi of H by a cluster Ai and connect some pairs of vertices of Ai and Aj if (vi,vj) is an edge of the graph H. As usual, we define the edge density between Ai and Aj as We study the following problem. Given densities γij for each edge (i,j) ∈ E(H), one has to decide whether there exists a blow-up graph G[H], with edge densities at least γij, such that one cannot choose a vertex from each cluster, so that the obtained graph is isomorphic to H, i.e., no H appears as a transversal in G[H]. We call dcrit(H) the maximal value for which there exists a blow-up graph G[H] with edge densities d(Ai,Aj)=dcrit(H) ((vi,vj) ∈ E(H)) not containing H in the above sense. Our main goal is to determine the critical edge density and to characterize the extremal graphs.

First, in the case of tree T we give an efficient algorithm to decide whether a given set of edge densities ensures the existence of a transversal T in the blow-up graph. Then we give general bounds on dcrit(H) in terms of the maximal degree. In connection with the extremal structure, the so-called star decomposition is proved to give the best construction for H-transversal-free blow-up graphs for several graph classes. Our approach applies algebraic graph-theoretical, combinatorial and probabilistic tools.

Let 1 ≤ p ≤ r + 1, with r ≥ 2 an integer, and let G be a graph of order n. Let d(v) denote the degree of a vertex v ∈ V(G). We show that if then G has more than (r + 1)-cliques sharing a common edge. From this we deduce that if then G contains more than cliques of order r + 1.

In turn, this statement is used to strengthen the Erdős–Stone theorem by using ∑v ∈ V(G)dp(v) instead of the number of edges.

We consider random permutations derived by sampling from stick-breaking partitions of the unit interval. The cycle structure of such a permutation can be associated with the path of a decreasing Markov chain on n integers. Under certain assumptions on the stick-breaking factor we prove a central limit theorem for the logarithm of the order of the permutation, thus extending the classical Erdős–Turán law for the uniform permutations and its generalization for Ewens' permutations associated with sampling from the PD/GEM(θ)-distribution. Our approach is based on using perturbed random walks to obtain the limit laws for the sum of logarithms of the cycle lengths.

We prove that the threshold for the appearance of a k-regular subgraph in Gn,p is at most the threshold for the appearance of a non-empty (k+1)-core. This improves a result of Pralat, Verstraete and Wormald [5] and proves a conjecture of Bollobás, Kim and Verstraete [3].

A spanning tree T of a graph G is called a homeomorphically irreducible spanning tree (HIST) if T does not contain vertices of degree 2. A graph G is called locally connected if, for every vertex v ∈ V(G), the subgraph induced by the neighbourhood of v is connected. In this paper, we prove that every connected and locally connected graph with more than 3 vertices contains a HIST. Consequently, we confirm the following conjecture due to Archdeacon: every graph that triangulates some surface has a HIST, which was proposed as a question by Albertson, Berman, Hutchinson and Thomassen.

More than forty years ago, Erdős conjectured that for any , every k-uniform hypergraph on n vertices without t disjoint edges has at most max edges. Although this appears to be a basic instance of the hypergraph Turán problem (with a t-edge matching as the excluded hypergraph), progress on this question has remained elusive. In this paper, we verify this conjecture for all . This improves upon the best previously known range , which dates back to the 1970s.

In this paper we analyse classical Maker–Breaker games played on the edge set of a sparse random board G ~ n,p. We consider the Hamiltonicity game, the perfect matching game and the k-connectivity game. We prove that for p(n) ≥ polylog(n)/n the board G ~ n,p is typically such that Maker can win these games asymptotically as fast as possible, i.e., within n+o(n), n/2+o(n) and kn/2+o(n) moves respectively.

A graph H is called common if the sum of the number of copies of H in a graph G and the number in the complement of G is asymptotically minimized by taking G to be a random graph. Extending a conjecture of Erdős, Burr and Rosta conjectured that every graph is common. Thomason disproved both conjectures by showing that K4 is not common. It is now known that in fact the common graphs are very rare. Answering a question of Sidorenko and of Jagger, Št'ovíček and Thomason from 1996 we show that the 5-wheel is common. This provides the first example of a common graph that is not three-colourable.

We prove the following metric Ramsey theorem. For any connected graph G endowed with a linear order on its vertex set, there exists a graph R such that in every colouring of the t-sets of vertices of R it is possible to find a copy G* of G inside R satisfying:

• • distG*(x, y) = distR(x, y) for every x, y ∈ V(G*);

• • the colour of each t-set in G* depends only on the graph-distance metric induced in G by the ordered t-set.

This paper deals with the partition function of the Ising model from statistical mechanics, which is used to study phase transitions in physical systems. A special case of interest is that of the Ising model with constant energies and external field. One may consider such an Ising system as a simple graph together with vertex and edge weights. When these weights are considered indeterminates, the partition function for the constant case is a trivariate polynomial Z(G;x,y,z). This polynomial was studied with respect to its approximability by Goldberg, Jerrum and Paterson. Z(G;x,y,z) generalizes a bivariate polynomial Z(G;t,y), which was studied in by Andrén and Markström.

We consider the complexity of Z(Gt,y) and Z(G;x,y,z) in comparison to that of the Tutte polynomial, which is well known to be closely related to the Potts model in the absence of an external field. We show that Z(G;x,y,z) is #P-hard to evaluate at all points in 3, except those in an exceptional set of low dimension, even when restricted to simple graphs which are bipartite and planar. A counting version of the Exponential Time Hypothesis, #ETH, was introduced by Dell, Husfeldt and Wahlén in order to study the complexity of the Tutte polynomial. In analogy to their results, we give under #ETH a dichotomy theorem stating that evaluations of Z(G;t,y) either take exponential time in the number of vertices of G to compute, or can be done in polynomial time. Finally, we give an algorithm for computing Z(G;x,y,z) in polynomial time on graphs of bounded clique-width, which is not known in the case of the Tutte polynomial.

Given a finite subset A of an abelian group G, we study the set k ∧ A of all sums of k distinct elements of A. In this paper, we prove that |k ∧ A| ≥ |A| for all k ∈ {2,. . .,|A| − 2}, unless k ∈ {2, |A| − 2} and A is a coset of an elementary 2-subgroup of G. Furthermore, we characterize those finite sets A ⊆ G for which |k ∧ A| = |A| for some k ∈ {2,. . .,|A| − 2}. This result answers a question of Diderrich. Our proof relies on an elementary property of proper edge-colourings of the complete graph.

A subgraph of a hypergraph H is even if all its degrees are positive even integers, and b-bounded if it has maximum degree at most b. Let fb(n) denote the maximum number of edges in a linearn-vertex 3-uniform hypergraph which does not contain a b-bounded even subgraph. In this paper, we show that if b ≥ 12, then for some absolute constant B, thus establishing fb(n) up to polylogarithmic factors. This leaves open the interesting case b = 2, which is the case of 2-regular subgraphs. We are able to show for some constants c, C > 0 that We conjecture that f2(n) = n1 + o(1) as n → ∞.

Let A and B be two affinely generating sets of ℤ2n. As usual, we denote their Minkowski sum by A+B. How small can A+B be, given the cardinalities of A and B? We give a tight answer to this question. Our bound is attained when both A and B are unions of cosets of a certain subgroup of ℤ2n. These cosets are arranged as Hamming balls, the smaller of which has radius 1.

By similar methods, we re-prove the Freiman–Ruzsa theorem in ℤ2n, with an optimal upper bound. Denote by F(K) the maximal spanning constant |〈 A 〉|/|A| over all subsets A ⊆ ℤ2n with doubling constant |A+A|/|A| ≤ K. We explicitly calculate F(K), and in particular show that 4K/4K ≤ F(K)⋅(1+o(1)) ≤ 4K/2K. This improves the estimate F(K) = poly(K)4K, found recently by Green and Tao [17] and by Konyagin [23].

Let F3,3 be the 3-graph on 6 vertices, labelled abcxyz, and 10 edges, one of which is abc, and the other 9 of which are all triples that contain 1 vertex from abc and 2 vertices from xyz. We show that for all n ≥ 6, the maximum number of edges in an F3,3-free 3-graph on n vertices is . This sharpens results of Zhou [9] and of the second author and Rödl [7].

Any function F: {0,. . ., N − 1} → {−1,1} such that F(x) can be computed from the binary digits of x using a bounded depth circuit is orthogonal to the Möbius function μ in the sense that

\[ \frac{1}{N} \sum_{0 \leq x \leq N-1} \mu(x)F(x) → 0 \quad\text{as}~~ N → \infty. \]

Consider the random graph process where we start with an empty graph on n vertices and, at time t, are given an edge et chosen uniformly at random among the edges which have not appeared so far. A classical result in random graph theory asserts that w.h.p. the graph becomes Hamiltonian at time (1/2+o(1))n log n. On the contrary, if all the edges were directed randomly, then the graph would have a directed Hamilton cycle w.h.p. only at time (1+o(1))n log n. In this paper we further study the directed case, and ask whether it is essential to have twice as many edges compared to the undirected case. More precisely, we ask if, at time t, instead of a random direction one is allowed to choose the orientation of et, then whether or not it is possible to make the resulting directed graph Hamiltonian at time earlier than n log n. The main result of our paper answers this question in the strongest possible way, by asserting that one can orient the edges on-line so that w.h.p. the resulting graph has a directed Hamilton cycle exactly at the time at which the underlying graph is Hamiltonian.

A graph is claw-free if it does not contain an induced subgraph isomorphic to K1,3. Cycles in claw-free graphs have been well studied. In this paper we extend results on disjoint cycles in claw-free graphs satisfying certain minimum degree conditions. In particular, we prove that if G is claw-free of sufficiently large order n = 3k with δ(G) ≥ n/2, then G contains k disjoint triangles.

Let kr(n, δ) be the minimum number of r-cliques in graphs with n vertices and minimum degree at least δ. We evaluate kr(n, δ) for δ ≤ 4n/5 and some other cases. Moreover, we give a construction which we conjecture to give all extremal graphs (subject to certain conditions on n, δ and r).

We show that the number of unit distances determined by n points in ℝ3 is O(n3/2), slightly improving the bound of Clarkson, Edelsbrunner, Guibas, Sharir and Welzl [5], established in 1990. The new proof uses the recently introduced polynomial partitioning technique of Guth and Katz [12]. While this paper was still in a draft stage, a similar proof of our main result was posted to the arXiv by Joshua Zahl [28].