Let n ≥ 3 be an integer, let V n (2) denote the vector space of dimension n over G F(2), and let c be the least residue of n modulo 3. We prove that the maximum number of 3-dimensional subspaces in V n (2) with pairwise intersection {0} is 2 n −2 c 7 − c for n ≥ 8 and c = 2. (The cases c = 0 and c = 1 have already been settled.) We then use our results to construct new optimal orthogonal arrays and (s, k, λ)-nets.

Abstract. Let n and t be positive integers with t < n, and let q be a prime power. A partial (t − 1)-spread of PG(n − 1, q) is a set of (t − 1)-dimensional subspaces of PG(n − 1, q) that are pairwise disjoint. Let r = n mod t and 0 ≤ r < t. We prove that if t > (q r − 1)/(q − 1), then the maximum size, i.e., cardinality, of a partial (t − 1)-spread of PG(n − 1, q) is (q n − q t+r )/(q t − 1) + 1. This essentially settles a main open problem in this area. Prior to this result, this maximum size was only known for r ∈ {0, 1} and for r = q = 2.

Let V n (q) denote a vector space of dimension n over the field with q elements. A set P of subspaces of V n (q) is a partition of V n (q) if every nonzero element of V n (q) is contained in exactly one element of P. Suppose there exists a partition of V n (q) into x i subspaces of dimension n i , 1 ≤ i ≤ k. Then x 1 , . . . , x k satisfy the Diophantine equation(q n i − 1)x i = q n − 1. However, not every solution of the Diophantine equation corresponds to a partition of V n (q). In this article, we show that there exists a partition of V n (2) into x subspaces of dimension 3 and y subspaces of dimension 2 if and only if 7x + 3y = 2 n − 1 and y = 1. In doing so, we introduce techniques useful in constructing further partitions. We also show that partitions of V n (q) induce uniformly resolvable designs on q n points.

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The generalized Turán number ex(G, H ) of two graphs G and H is the maximum number of edges in a subgraph of G not containing H . When G is the complete graph K m on m vertices, the value of ex(K m , H )where o(1) → 0 as m → ∞, by the Erdős-Stone-Simonovits theorem. In this paper we give an analogous result for triangle-free graphs H and pseudo-random graphs G.