A coloring problem for intersecting vector spaces

In Hoppen et al. (2012) Kohayakawa and two of the current authors considered a variant of the classical Erdős–Ko–Rado problem for families of ℓℓ-intersecting rr-sets in which they asked for the maximum number of edge-colorings of an nn-vertex rr-uniform hypergraph such that all color classes are ℓℓ-intersecting. This resulted in a fairly complete characterization of the corresponding extremal families. In this paper, we show that, when the number of colors is k∈{2,3,4}k∈{2,3,4}, similar results may be obtained in the context of vector spaces. In particular, we observe that a rather unusual instability phenomenon occurs for k=4k=4 colors, namely that the problem is unstable despite admitting a unique extremal configuration up to isomorphism.