A q-ANALOGUE OF A PROBLEM OF ERDŐS AND ROTHSCHILD

Erdős and Rothschild [5] proposed a variant of the classical Turán problem in which they asked for the maximum number of edge-colorings of an n-vertex graph avoiding monochromatic copies of some given forbidden subgraph. The same variant has been addressed in [11] in connection with the Erdős–Ko–Rado Theorem for families of `-intersecting r-sets, which 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}, these results can be translated to the context of vector spaces. In particular, we observe that a rather unusual instability phenomenon occurs for k = 4 colors, namely that the problem is unstable despite admitting a unique extremal configuration up to isomorphism.