Intersection theorems under dimension constraints

In Ahlswede et al. [Discrete Math. 273(1–3) (2003) 9–21] we posed a series of extremal (set system) problems under dimension constraints. In the present paper, we study one of them: the intersection problem. The geometrical formulation of our problem is as follows. Given integers 0 ⩽ t , k ⩽ n determine or estimate the maximum number of ( 0 , 1 ) -vectors in a k-dimensional subspace of the Euclidean n-space R n , such that the inner product (“intersection”) of any two is at least t. Also we are interested in the restricted (or the uniform) case of the problem; namely, the problem considered for the ( 0 , 1 ) -vectors of the same weight ω . per consists of two parts, which concern similar questions but are essentially independent with respect to the methods used. t I, we consider the unrestricted case of the problem. Surprisingly, in this case the problem can be reduced to a weighted version of the intersection problem for systems of finite sets. A general conjecture for this problem is proved for the cases mentioned in Ahlswede et al. [Discrete Math. 273(1–3) (2003) 9–21]. We also consider a diametric problem under dimension constraint. t II, we study the restricted case and solve the problem for t = 1 and k < 2 ω , and also for any fixed 1 ⩽ t ⩽ ω and k large.