This special issue is devoted to papers from the meeting on Combinatorics, Probability and Computing, held at the Mathematisches Forschungsinstitut in Oberwolfach from 29 October 2006 to 4 November 2006. Like most conferences at Oberwolfach, this was an exciting and stimulating occasion; we were treated to numerous excellent talks, many of which provoked a great deal of interest and discussion among the participants.

Ingo Wegener passed away on 26 November 2008, after a three-year struggle with brain cancer. This is a tremendous loss for all those who knew him and worked with him. Ingo is survived by his wife, Christa.

After studying Mathematics in Bielefeld and holding a post in Frankfurt am Main, in 1987 Ingo became a full professor of Computer Science, in Efficient Algorithms and Complexity Theory, at the Technische Universität Dortmund, a position he held until his death.

We describe a lower bound for the rank of any real matrix in which all diagonal entries are significantly larger in absolute value than all other entries, and discuss several applications of this result to the study of problems in Geometry, Coding Theory, Extremal Finite Set Theory and Probability. This is partly a survey, containing a unified approach for proving various known results, but it contains several new results as well.

In majority bootstrap percolation on a graph G, an infection spreads according to the following deterministic rule: if at least half of the neighbours of a vertex v are already infected, then v is also infected, and infected vertices remain infected forever. We say that percolation occurs if eventually every vertex is infected.

The elements of the set of initially infected vertices, A ⊂ V(G), are normally chosen independently at random, each with probability p, say. This process has been extensively studied on the sequence of torus graphs [n]d, for n = 1,2, . . ., where d = d(n) is either fixed or a very slowly growing function of n. For example, Cerf and Manzo [17] showed that the critical probability is o(1) if d(n) ≤ log*n, i.e., if p = p(n) is bounded away from zero then the probability of percolation on [n]d tends to one as n → ∞.

In this paper we study the case when the growth of d to ∞ is not excessively slow; in particular, we show that the critical probability is 1/2 + o(1) if d ≥ (log log n)2 log log log n, and give much stronger bounds in the case that G is the hypercube, [2]d.

We study relaxations of proper two-colourings, such that the order of the induced monochromatic components in one (or both) of the colour classes is bounded by a constant. A colouring of a graph G is called (C1, C2)-relaxed if every monochromatic component induced by vertices of the first (second) colour is of order at most C1 (C2, resp.). We prove that the decision problem ‘Is there a (1, C)-relaxed colouring of a given graph G of maximum degree 3?’ exhibits a hardness jump in the component order C. In other words, there exists an integer f(3) such that the decision problem is NP-hard for every 2 ≤ C < f(3), while every graph of maximum degree 3 is (1, f(3))-relaxed colourable. We also show f(3) ≤ 22 by way of a quasilinear time algorithm, which finds a (1, 22)-relaxed colouring of any graph of maximum degree 3. Both the bound on f(3) and the running time greatly improve earlier results. We also study the symmetric version, that is, when C1 = C2, of the relaxed colouring problem and make the first steps towards establishing a similar hardness jump.

Given ω ≥ 1, let be the graph with vertex set in which two vertices are joined if they agree in one coordinate and differ by at most ω in the other. (Thus is precisely .) Let pc(ω) be the critical probability for site percolation on . Extending recent results of Frieze, Kleinberg, Ravi and Debany, we show that limω→∞ωpc(ω)=log(3/2). We also prove analogues of this result for the n-by-n grid and in higher dimensions, the latter involving interesting connections to Gilbert's continuum percolation model. To prove our results, we explore the component of the origin in a certain non-standard way, and show that this exploration is well approximated by a certain branching random walk.

A family of subsets of {1, . . ., n} is called a j-junta if there exists J ⊆ {1, . . ., n}, with |J| = j, such that the membership of a set S in depends only on S ∩ J.

In this paper we provide a simple description of intersecting families of sets. Let n and k be positive integers with k < n/2, and let be a family of pairwise intersecting subsets of {1, . . ., n}, all of size k. We show that such a family is essentially contained in a j-junta , where j does not depend on n but only on the ratio k/n and on the interpretation of ‘essentially’.

When k = o(n) we prove that every intersecting family of k-sets is almost contained in a dictatorship, a 1-junta (which by the Erdős–Ko–Rado theorem is a maximal intersecting family): for any such intersecting family there exists an element i ∈ {1, . . ., n} such that the number of sets in that do not contain i is of order (which is approximately times the size of a maximal intersecting family).

Our methods combine traditional combinatorics with results stemming from the theory of Boolean functions and discrete Fourier analysis.

Jim Propp's rotor–router model is a deterministic analogue of a random walk on a graph. Instead of distributing chips randomly, each vertex serves its neighbours in a fixed order. We analyse the difference between the Propp machine and random walk on the infinite two-dimensional grid. It is known that, apart from a technicality, independent of the starting configuration, at each time the number of chips on each vertex in the Propp model deviates from the expected number of chips in the random walk model by at most a constant. We show that this constant is approximately 7.8 if all vertices serve their neighbours in clockwise or order, and 7.3 otherwise. This result in particular shows that the order in which the neighbours are served makes a difference. Our analysis also reveals a number of further unexpected properties of the two-dimensional Propp machine.

Random geometric graphs have been one of the fundamental models for reasoning about wireless networks: one places n points at random in a region of the plane (typically a square or circle), and then connects pairs of points by an edge if they are within a fixed distance of one another. In addition to giving rise to a range of basic theoretical questions, this class of random graphs has been a central analytical tool in the wireless networking community.

For many of the primary applications of wireless networks, however, the underlying environment has a large number of obstacles, and communication can only take place among nodes when they are close in space and when they have line-of-sight access to one another – consider, for example, urban settings or large indoor environments. In such domains, the standard model of random geometric graphs is not a good approximation of the true constraints, since it is not designed to capture the line-of-sight restrictions.

Here we propose a random-graph model incorporating both range limitations and line-of-sight constraints, and we prove asymptotically tight results for k-connectivity. Specifically, we consider points placed randomly on a grid (or torus), such that each node can see up to a fixed distance along the row and column it belongs to. (We think of the rows and columns as ‘streets’ and ‘avenues’ among a regularly spaced array of obstructions.) Further, we show that when the probability of node placement is a constant factor larger than the threshold for connectivity, near-shortest paths between pairs of nodes can be found, with high probability, by an algorithm using only local information. In addition to analysing connectivity and k-connectivity, we also study the emergence of a giant component, as well an approximation question, in which we seek to connect a set of given nodes in such an environment by adding a small set of additional ‘relay’ nodes.

Let C(3)n denote the 3-uniform tight cycle, that is, the hypergraph with vertices v1, .–.–., vn and edges v1v2v3, v2v3v4, .–.–., vn−1vnv1, vnv1v2. We prove that the smallest integer N = N(n) for which every red–blue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C(3)n is asymptotically equal to 4n/3 if n is divisible by 3, and 2n otherwise. The proof uses the regularity lemma for hypergraphs of Frankl and Rödl.

Consider a random multigraph G* with given vertex degrees d1,. . .,dn, constructed by the configuration model. We show that, asymptotically for a sequence of such multigraphs with the number of edges , the probability that the multigraph is simple stays away from 0 if and only if . This was previously known only under extra assumptions on the maximum degree maxidi. We also give an asymptotic formula for this probability, extending previous results by several authors.

This paper has two main focal points. We first consider an important class of machine learning algorithms: large margin classifiers, such as Support Vector Machines. The notion of margin complexity quantifies the extent to which a given class of functions can be learned by large margin classifiers. We prove that up to a small multiplicative constant, margin complexity is equal to the inverse of discrepancy. This establishes a strong tie between seemingly very different notions from two distinct areas.

In the same way that matrix rigidity is related to rank, we introduce the notion of rigidity of margin complexity. We prove that sign matrices with small margin complexity rigidity are very rare. This leads to the question of proving lower bounds on the rigidity of margin complexity. Quite surprisingly, this question turns out to be closely related to basic open problems in communication complexity, e.g., whether PSPACE can be separated from the polynomial hierarchy in communication complexity.

Communication is a key ingredient in many types of learning. This explains the relations between the field of learning theory and that of communication complexity [6, l0, 16, 26]. The results of this paper constitute another link in this rich web of relations. These new results have already been applied toward the solution of several open problems in communication complexity [18, 20, 29].

For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum n such that every edge colouring of the complete graph on n vertices (with any number of colours) has a monochromatic subgraph isomorphic to S or a rainbow subgraph isomorphic to T. Here, a subgraph is said to be rainbow if all of its edges have different colours. It is an immediate consequence of the Erdős–Rado Canonical Ramsey Theorem that f(S, T) exists if and only if S is a star or T is acyclic. Much work has been done to determine the rate of growth of f(S, T) for various types of parameters. When S and T are both trees having s and t edges respectively, Jamison, Jiang and Ling showed that f(S, T) ≤ O(st2) and conjectured that it is always at most O(st). They also mentioned that one of the most interesting open special cases is when T is a path. In this paper, we study this case and show that f(S, Pt) = O(st log t), which differs only by a logarithmic factor from the conjecture. This substantially improves the previous bounds for most values of s and t.

Consider the following one-player game. Starting with the empty graph on n vertices, in every step a new edge is drawn uniformly at random and inserted into the current graph. This edge has to be coloured immediately with one of r available colours. The player's goal is to avoid creating a monochromatic copy of some fixed graph F for as long as possible. We prove an upper bound on the typical duration of this game if F is from a large class of graphs including cliques and cycles of arbitrary size. Together with lower bounds published elsewhere, explicit threshold functions follow.

Consider the following one-player game. Starting with the empty graph on n vertices, in every step a new edge is drawn uniformly at random and inserted into the current graph. This edge has to be coloured immediately with one of r available colours. The player's goal is to avoid creating a monochromatic copy of some fixed graph F for as long as possible. We prove a lower bound of nβ(F,r) on the typical duration of this game, where β(F,r) is a function that is strictly increasing in r and satisfies limr→∞ β(F,r) = 2 − 1/m2(F), where n2−1/m2(F) is the threshold of the corresponding offline colouring problem.