Let M(n, A) denote the maximum possible cardinality of a family of binary strings of length n, such that for every four distinct members of the family there is a coordinate in which exactly two of them have a 1. We prove that M(n, A) [les ] 20.78n for all sufficiently large n. Let M(n, C) denote the maximum possible cardinality of a family of binary strings of length n, such that for every four distinct members of the family there is a coordinate in which exactly one of them has a 1. We show that there is an absolute constant c < 1/2 such that M(n, C) [les ] 2cn for all sufficiently large n. Some related questions are discussed as well.

Vapnik and Chervonenkis proposed in [7] a combinatorial notion of dimension that reflects the ‘combinatorial complexity’ of families of sets. In the three decades that have passed since that paper, this notion – the Vapnik–Chervonenkis dimension (VC-dimension) – has been discovered to be of primal importance in quite a wide variety of topics in both pure mathematics and theoretical computer science.

In this paper we turn our attention to classes with infinite VC-dimension, a realm thrown into one big bag by the usual VC-dimension analysis. We identify three levels of combinatorial complexity of classes with infinite VC-dimension. We show that these levels fall under the set-theoretic definition of σ-ideals (in particular, each of them is closed under countable unions), and that they are all distinct. The first of these levels (i.e., the family of ‘small’ infinite-dimensional classes) coincides with the family of classes which are non-uniformly PAC-learnable.

Maybe the most surprising contribution of this work is the discovery of an intimate relation between the VC-dimension of a class of subsets of the natural numbers and the Lebesgue measure of the set of reals defined when these subsets are viewed as binary representations of real numbers.

As a by-product, our investigation of the VC-dimension-induced ideals over the reals yields a new proper extension of the Lebesgue measure. Another offshoot of this work is a simple result in probability theory, showing that, given any sequence of pairwise independent events, any random event is eventually independent of the members of the sequence.

We examine the specialization to simple matroids of certain problems in extremal matroid theory that are concerned with bounded cocircuit size. Assume that each cocircuit of a simple matroid M has at most d elements. We show that if M has rank 3, then M has at most d + [lfloor ]√d[rfloor ] + 1 points, and we classify the rank-3 simple matroids M that have exactly d + [lfloor ]√d[rfloor ] points. We show that if M is a connected matroid of rank 4 and d is q3 with q > 1, then M has at most q3 + q2 + q + 1 points; this upper bound is strict unless q is a prime power, in which case the only such matroid with exactly q3 + q2 + q + 1 points is the projective geometry PG(3, q). We also show that if d is q4 for a positive integer q and if M has rank 5 and is vertically 5-connected, then M has at most q4 + q3 + q2 + q + 1 points; this upper bound is strict unless q is a prime power, in which case PG(4, q) is the only such matroid that attains this bound.

Consider the integer lattice L = ℤ2. For some m [ges ] 4, let us colour each column of this lattice independently and uniformly with one of m colours. We do the same for the rows, independently of the columns. A point of L will be called blocked if its row and column have the same colour. We say that this random configuration percolates if there is a path in L starting at the origin, consisting of rightward and upward unit steps, avoiding the blocked points. As a problem arising in distributed computing, it has been conjectured that for m [ges ] 4 the configuration percolates with positive probability. This question remains open, but we prove that the probability that there is percolation to distance n but not to infinity is not exponentially small in n. This narrows the range of methods available for proving the conjecture.

In this paper, we combine two previous works, the first being by the first author and K. Nelander, and the second by J. van den Berg and the second author, to show (1) that one can carry out a Propp–Wilson exact simulation for all Markov random fields on ℤd satisfying a certain high noise assumption, and (2) that all such random fields are a finitary image of a finite state i.i.d. process. (2) is a strengthening of the previously known fact that such random fields are so-called Bernoulli shifts.

The intersection exponent ξ for simple random walk in two and three dimensions gives a measure of the rate of decay of the probability that paths do not intersect. In this paper we show that the intersection exponent for random walks is the same as that for Brownian motion and show in fact that the probability of nonintersection up to distance n is comparable (equal up to multiplicative constants) to n−ξ.

We derive improved isoperimetric inequalities for discrete product measures on the n-dimensional cube. As a consequence, a general theorem on the threshold behaviour of monotone properties is obtained. This is then applied to coding theory when we study the probability of error after decoding.