Intersection theorems for systems of finite vector spaces

Abstract

A theorem of Erdös, Ko and Rado states that if S is an n-element set and F is a family of k-element subsets of S, k⩽ 1 2 n , such that no two members of F are disjoint, then … F… ⩽ ( n - 1 k - 1) . In this paper we investigate the analogous problem for finite vector spaces. Let F be a family of k-dimensional subspaces of an n-dimensional vector space over a field of q elements such that members of F intersect pairwise non-trivially. Employing a method of Katona, we show that for n ⩾ 2k, … F… ⩽ (k/n) [ n k]q . By a more detailed analysis, we obtain that for n ⩾ 2k + 1, … F… ⩽ [ n - 1 k - 1]q , which is a best possible bound. The argument employed is generalized to the problem of finding a bound on the size of F when its members have pairwise intersection dimension no smaller than r. Again best possible results are obtained for n ⩾ 2 k + 2 and n ⩾ 2 k + 1, q ⩾ 3. Application of these methods to the analogous subset problem leads to improvement on the Erdös-Ko-Rado bounds.