Partition theorems for subspaces of vector spaces

Abstract

The principal result of this paper provides a nearly complete answer to the following question. For which cardinal numbers t , m , n , q and r is it true that whenever the t -dimensional subspaces of an n -dimensional vector space V over a field of q elements are partitioned into r classes, there must be some m -dimensional subspace of V, all of whose t -dimensional subspaces lie in the same class? This question is answered completely if r ≤ N 0. The contributions of this paper are in the form of negative answers, since it turns out that all affirmative answers (which we have) were already known or easily deducible from known results.