A redundant version of the Rado–Horn Theorem

Journal of Combinatorial Designs, 2008

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 k i=1 (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.

Linear Algebra and its Applications, 2003

Let V , W be finite dimensional vector spaces over a field K, each with n distinguished subspaces, with a dimension-preserving correspondence between intersections. When does this guarantee the existence of an isomorphism between V and W matching corresponding subspaces? The setting where it happens requires that the distinguished subspaces be generated by subsets of a given redundant base of the space; this gives rise to a (0,1)-incidence table called tent, an object which occurs in the study of Butler B(1)-groups.

Let V be an n-dimensional vector space over a finite field. Assign a real-valued weight to each 1-dimensional subspace in V so that the sum of all weights is zero. Define the weight of a subspace S ⊂ V to be the sum of the weights of all the 1-dimensional subspaces it contains. We prove that if n ≥ 3k, then the number of k-dimensional subspaces in V with nonnegative weight is at least the number of k-dimensional subspaces in V that contain a fixed 1-dimensional subspace. This result verifies a conjecture of Manickam and Singhi from 1988.

Journal of Combinatorial Theory, Series A, 1994

It is shown that the convex hull of a spanning set V of L½(d+ 1)kJ + 1 points in Nd has a supporting spanned hyperplane (equivalently, a facet of [V]) which misses at least k points of V. Spanning sets of cardinality L½(d+ 1)kj not having such a supporting hyperplane are fully described. The related problem where the spanned hyperplane does not have to be supporting, but still leaves at least k points of V on one side, is discussed and solved in some instances (e.g., d odd or k ~< 9, k ~ 8).

IEEE Transactions on Information Theory, 2005

Two subspaces of a vector space are here called "nonintersecting" if they meet only in the zero vector. Motivated by the design of noncoherent multiple-antenna communications systems, we consider the following question. How many pairwise nonintersecting M t-dimensional subspaces of an m-dimensional vector space V over a field F can be found, if the generator matrices for the subspaces may contain only symbols from a given finite alphabet A ⊆ F? The most important case is when F is the field of complex numbers C ; then M t is the number of antennas. If A = F = GF (q) it is shown * This work was carried out during F. E. Oggier's visit to AT&T Shannon Labs during the summer of 2003. She thanks the Fonds National Suisse, Bourses et Programmes d'Échange for support. case when F = C only the case M t = 2 is considered. It is shown that if A is a PSK-configuration, consisting of the 2 r complex roots of unity, the number of nonintersecting planes is at least 2 r(m−2) and at most 2 r(m−1)−1 (the lower bound may in fact be the best that can be achieved).

2015

For a vector space $V$ the \emph{intersection graph of subspaces} of $V$, denoted by $G(V)$, is the graph whose vertices are in a one-to-one correspondence with proper nontrivial subspaces of $V$ and two distinct vertices are adjacent if and only if the corresponding subspaces of $V$ have a nontrivial (nonzero) intersection. In this paper, we study the clique number, the chromatic number, the domination number and the independence number of the intersection graphs of subspaces of a vector space.

Linear and Multilinear Algebra, 2018

Motivated by existence problems of the maximum number of spanning sets (MS) and the minimum number of Linearly independent sets (ML) which partition a generator in a finite dimensional vector space, we introduce a real time iterative algorithm with polynomial complexity. The algorithm starts with an arbitrary partition and improves it to obtain the final partition. In addition this algorithm gives the conclusion of the Rado-Horn theorem, and facilitates using of this theorem.

Integers, 2000

We prove that if a subset of a d-dimensional vector space over a finite field with q elements has more than q d−1 elements, then it determines all the possible directions. If a set has more than q k elements, it determines a k-dimensional set of directions. We prove stronger results for sets that are sufficiently random. This result is best possible as the example of a k-dimensional hyperplane shows. We can view this question as an Erdős type problem where a sufficiently large subset of a vector space determines a large number of configurations of a given type. See, for example, ([7]), [1], [4], [11], [12] and the references contained therein. For discrete subsets of R d , this question has been previously studied by Pach, Pinchasi and Sharir. See ([8]).

Designs, Codes and Cryptography, 2023

In this paper we settle the question of whether a finite-dimensional vector space V over F p , with p an odd prime, and the family of all the k-sets of elements of V summing up to a given element x, form a 1-(v, k, λ 1 ) or a 2-(v, k, λ 2 ) block design, and, in either case, we find a closed form for λ i and characterize the automorphism group. The question is discussed also in the case where the elements of the k-sets are required to be all nonzero, as the two cases happen to be intrinsically inseparable. The "twin case" p = 2, which has strict connections with coding theory, was completely discussed in a recent paper by G. Falcone and the present author.

2011

For a vector space $V$ the \emph{intersection graph of subspaces} of $V$, denoted by $G(V)$, is the graph whose vertices are in a one-to-one correspondence with proper nontrivial subspaces of $V$ and two distinct vertices are adjacent if and only if the corresponding subspaces of $V$ have a nontrivial (nonzero) intersection. In this paper, we study the clique number, the