A Colored Version of the Erdős-ko-rado Theorem for Vector Spaces

We study a variant, in the context of the Erdős–Ko–Rado Theorem for vector spaces [4], of an extremal problem of Erdős and Rothschild [3], who considered edge-colorings of graphs avoiding monochromatic triangles. For fixed positive integers r, k and ` with 1 ≤ ` < r, and a family F of linear r-dimensional subspaces in a linear n-dimensional vector space Vn over the fixed finite field GF (q), let C(k,`)(F) denote the number of k-colorings of spaces in F for which the intersection of any two spaces in the same color class has dimension at least `. Consider the function χ(r,k,`,q)(n) = maxF⊆Sn,q,r C(k,`)(F), where the maximum runs over all subfamilies of the family Sn,q,r of all linear r-dimensional spaces in a linear n-dimensional space Vn over GF (q). In this paper, we study the asymptotic behavior of the function χ(r,k,`,q)(n) and the corresponding extremal families.