This paper is part of the ongoing effort to study high-dimensional permutations. We prove the analogue to the Erdős–Szekeres theorem: For every k ≥ 1, every order-nk-dimensional permutation contains a monotone subsequence of length Ωk ( $\sqrt{n}$ ), and this is tight. On the other hand, and unlike the classical case, the longest monotone subsequence in a random k-dimensional permutation of order n is asymptotically almost surely Θk (n k/(k+1)).

Assume that the edges of the complete graph Kn are given independent uniform [0, 1] weights. We consider the expected minimum total weight μk of k ⩽ 2 edge-disjoint spanning trees. When k is large we show that μk ≈ k 2. Most of the paper is concerned with the case k = 2. We show that m 2 tends to an explicitly defined constant and that μ 2 ≈ 4.1704288. . . .

We prove that there is a set F in the plane so that the distance between any two points of F is at least 1, and for any positive ϵ < 1, and every line segment in the plane of length at least ϵ−1−o(1), there is a point of F within distance ϵ of the segment. This is tight up to the o(1)-term in the exponent, improving earlier estimates of Peres, of Solomon and Weiss, and of Adiceam.

This paper deals with a combinatorial problem concerning colourings of uniform hypergraphs with large girth. We prove that if H is an n-uniform non-r-colourable simple hypergraph then its maximum edge degree Δ(H) satisfies the inequality

$$\Delta(H)\geqslant c\cdot r^{n-1}\ffrac{n(\ln\ln n)^2}{\ln n}$$

As an application of our probabilistic technique we establish a lower bound for the classical van der Waerden number W(n, r), the minimum natural N such that in an arbitrary colouring of the set of integers {1,. . .,N} with r colours there exists a monochromatic arithmetic progression of length n. We prove that

$$W(n,r)\geqslant c\cdot r^{n-1}\ffrac{(\ln\ln n)^2}{\ln n}.$$

Finding a hidden partition in a random environment is a general and important problem which contains as subproblems many important questions, such as finding a hidden clique, finding a hidden colouring, finding a hidden bipartition, etc.

In this paper we provide a simple SVD algorithm for this purpose, addressing a question of McSherry. This algorithm is easy to implement and works for sparse graphs under optimal density assumptions. We also consider an approximating algorithm, which on one hand works under very mild assumptions, but on other hand can sometimes be upgraded to give the exact solution.

Bollobás and Scott (Random Struct. Alg.21 (2002) 414–430) asked for conditions that guarantee a bisection of a graph with m edges in which each class has at most (1/4+o(1))m edges. We demonstrate that cycles of length 4 play an important role for this question. Let G be a graph with m edges, minimum degree δ, and containing no cycle of length 4. We show that if (i) G is 2-connected, or (ii) δ ⩾ 3, or (iii) δ ⩾ 2 and the girth of G is at least 5, then G admits a bisection in which each class has at most (1/4+o(1))m edges. We show that each of these conditions are best possible. On the other hand, a construction by Alon, Bollobás, Krivelevich and Sudakov shows that for infinitely many m there exists a graph with m edges and girth at least 5 for which any bisection has at least (1/4−o(1))m edges in one of the two classes.

Due to a calculation error, the constant in the main theorem is not $80 \sqrt{k\log k}$ but $80\sqrt{k} \log k$ . The error was discovered by Xizhi Liu.

For a new discussion on the limit of the method used in this paper see the revised arXiv version of the paper.

A cutset is a non-empty finite subset of ℤd which is both connected and co-connected. A cutset is odd if its vertex boundary lies in the odd bipartition class of ℤd . Peled [18] suggested that the number of odd cutsets which contain the origin and have n boundary edges may be of order e Θ(n/d) as d → ∞, much smaller than the number of general cutsets, which was shown by Lebowitz and Mazel [15] to be of order d Θ(n/d). In this paper, we verify this by showing that the number of such odd cutsets is (2+o(1))n/2d .

Recently there has been much interest in studying random graph analogues of well-known classical results in extremal graph theory. Here we follow this trend and investigate the structure of triangle-free subgraphs of G(n, p) with high minimum degree. We prove that asymptotically almost surely each triangle-free spanning subgraph of G(n, p) with minimum degree at least (2/5 + o(1))pn is (p −1 n)-close to bipartite, and each spanning triangle-free subgraph of G(n, p) with minimum degree at least (1/3 + ϵ)pn is O(p −1 n)-close to r-partite for some r = r(ϵ). These are random graph analogues of a result by Andrásfai, Erdős and Sós (Discrete Math.8 (1974), 205–218), and a result by Thomassen (Combinatorica22 (2002), 591–596). We also show that our results are best possible up to a constant factor.

We almost completely solve a number of problems related to a concept called majority colouring recently studied by Kreutzer, Oum, Seymour, van der Zypen and Wood. They raised the problem of determining, for a natural number k, the smallest number m = m(k) such that every digraph can be coloured with m colours where each vertex has the same colour as at most a 1/k proportion of its out-neighbours. We show that m(k) ∈ {2k − 1,2k}. We also prove a result supporting the conjecture that m(2) = 3. Moreover, we prove similar results for a more general concept called majority choosability.

We consider the problem of minimizing the number of edges that are contained in triangles, among n-vertex graphs with a given number of edges. For sufficiently large n, we prove an exact formula for this minimum, which partially resolves a conjecture of Füredi and Maleki.

A sequence S is called anagram-free if it contains no consecutive symbols r1r2. . .rkrk+1. . .r2k such that rk+1. . .r2k is a permutation of the block r1r2. . .rk. Answering a question of Erdős and Brown, Keränen constructed an infinite anagram-free sequence on four symbols. Motivated by the work of Alon, Grytczuk, Hałuszczak and Riordan [2], we consider a natural generalization of anagram-free sequences for graph colourings. A colouring of the vertices of a given graph G is called anagram-free if the sequence of colours on any path in G is anagram-free. We call the minimal number of colours needed for such a colouring the anagram-chromatic number of G.

In this paper we study the anagram-chromatic number of several classes of graphs like trees, minor-free graphs and bounded-degree graphs. Surprisingly, we show that there are bounded-degree graphs (such as random regular graphs) in which anagrams cannot be avoided unless we essentially give each vertex a separate colour.

We consider the complete graph 𝜅n on n vertices with exponential mean n edge lengths. Writing Cij for the weight of the smallest-weight path between vertices i, j ∈ [n], Janson [18] showed that maxi,j∈[n]C ij/logn converges in probability to 3. We extend these results by showing that maxi,j∈[n]Cij − 3 logn converges in distribution to some limiting random variable that can be identified via a maximization procedure on a limiting infinite random structure. Interestingly, this limiting random variable has also appeared as the weak limit of the re-centred graph diameter of the barely supercritical Erdős–Rényi random graph in [22].

Given hypergraphs F and H, an F-factor in H is a set of vertex-disjoint copies of F which cover all the vertices in H. Let K − 4 denote the 3-uniform hypergraph with four vertices and three edges. We show that for sufficiently large n ∈ 4ℕ, every 3-uniform hypergraph H on n vertices with minimum codegree at least n/2−1 contains a K − 4-factor. Our bound on the minimum codegree here is best possible. It resolves a conjecture of Lo and Markström [15] for large hypergraphs, who earlier proved an asymptotically exact version of this result. Our proof makes use of the absorbing method as well as a result of Keevash and Mycroft [11] concerning almost perfect matchings in hypergraphs.

A subset C of edges in a k-uniform hypergraph H is a loose Hamilton cycle if C covers all the vertices of H and there exists a cyclic ordering of these vertices such that the edges in C are segments of that order and such that every two consecutive edges share exactly one vertex. The binomial random k-uniform hypergraph H k n,p has vertex set [n] and an edge set E obtained by adding each k-tuple e ∈ ( $\binom{[n]}{k}$ ) to E with probability p, independently at random.

Here we consider the problem of finding edge-disjoint loose Hamilton cycles covering all but o(|E|) edges, referred to as the packing problem. While it is known that the threshold probability of the appearance of a loose Hamilton cycle in H k n,p is

$p=\Theta\biggl(\frac{\log n}{n^{k-1}}\biggr),$

$N:=(1-o(1))\frac{\binom{n}{k}p}{n/(k-1)}$

Our proof utilizes and modifies the idea of ‘online sprinkling’ recently introduced by Vu and the first author.

In this paper we prove lower and upper bounds for the extinction time of the contact process on random geometric graphs with connection radius tending to infinity. We obtain that for any infection rate λ > 0, the contact process on these graphs survives a time super-exponential in the number of vertices.

We study factor of i.i.d. processes on the d-regular tree for d ≥ 3. We show that if such a process is restricted to two distant connected subgraphs of the tree, then the two parts are basically uncorrelated. More precisely, any functions of the two parts have correlation at most $k(d-1) / (\sqrt{d-1})^k$ , where k denotes the distance between the subgraphs. This result can be considered as a quantitative version of the fact that factor of i.i.d. processes have trivial 1-ended tails.

Let hom(G) denote the size of the largest clique or independent set of a graph G. In 2007, Bukh and Sudakov proved that every n-vertex graph G with hom(G) = O(logn) contains an induced subgraph with Ω(n 1/2) distinct degrees, and raised the question of deciding whether an analogous result holds for every n-vertex graph G with hom(G) = O(nϵ ), where ϵ > 0 is a fixed constant. Here, we answer their question in the affirmative and show that every graph G on n vertices contains an induced subgraph with Ω((n/hom(G))1/2) distinct degrees. We also prove a stronger result for graphs with large cliques or independent sets and show, for any fixed k ∈ ℕ, that if an n-vertex graph G contains no induced subgraph with k distinct degrees, then hom(G)⩾n/(k − 1) − o(n); this bound is essentially best possible.

In my paper [1], a normalization factor of |G|−1 was missing in the statements of Theorem 2.4, Theorem 2.6 and Corollary 2.7.

We identify a relationship between a certain family of random walks on Euclidean lattices and difference matrices over cyclic groups. We then use the techniques of Fourier analysis to estimate the return probabilities of these random walks, which in turn yields the asymptotic number of difference matrices over cyclic groups as the number of columns increases.