For every integer N, we explicitly construct a subset of residues mod N of size(log N)o(1) which is nearly uniformly distributed in every arithmetic progression modulo N.

We consider a simple randomised algorithm that seeks a weak 2-colouring of a hypergraph H; that is, it tries to 2-colour the points of H so that no edge is monochromatic. If H has a particular well-behaved form of such a colouring, then the method is successful within expected number of iterations O(n3) when H has n points. In particular, when applied to a graph G with n nodes and chromatic number 3, the method yields a 2-colouring of the vertices such that no triangle is monochromatic in expected time O(n4).

Two natural classes of polymatroids can be associated with hypergraphs: the so-called Boolean and hypergraphic polymatroids. Boolean polymatroids carry virtually all the structure of hypergraphs; hypergraphic polymatroids generalize graphic matroids. This paper considers algorithmic problems associated with recognizing members of these classes. Let k be a fixed positive integer and assume that the k-polymatroid ρ is presented via a rank oracle. We present an algorithm that determines in polynomial time whether ρ is Boolean, and if it is, finds the hypergraph. We also give an algorithm that decides in polynomial time whether ρ is the hypergraphic polymatroid associated with a given hypergraph. Other structure-theoretic results are also given.