On a generalization of linear independence in finite-dimensional vector spaces

AbstractA set X in a vector space V is said to be k-independent (where k is a positive integer) if, for each x ϵ X,X{x} admits a partition into k subsets {χgq}θ=1,…,k such that x ∉ span χθ, θ = 1,…,k. It is proved that if dim V = n and X ⊂ V is k-independent, then X cannot contain more than (n+k−1k) elements; this bound is sharp for vector spaces over sufficiently large fields. A broader notion (k-independence in degrees) is considered, and similar results are obtained. Several unresolved problems are stated, some involving matroid generalizations of questions answered as yet only within vector space context