Colour Schemes

If the k-subsets of a large set are coloured with r colours (all of which are used), then at least r distinct colour schemes of (k + 1)-sets occur. This paper considers the problem of determining colourings with equally many colours and colour schemes. Some general results (including a reduction to r = 2) are given, and the cases k = 2 and (k, r) = (3,2) are treated in detail. The problem arose from-a question about orbits of permutation groups on unordered sets; applications to permutation groups, and to enumerative model theory, are discussed. Some possible generalisations are mentioned.