On Non-Optimally Expanding Sets in Grassmann Graphs

The paper investigates expansion properties of the Grassmann graph, motivated by recent results of [KMS16, DKK16] concerning hardness of the Vertex-Cover and of the 2-to-1 Games problems. Proving the hypotheses put forward by these papers seems to first require a better understanding of these expansion properties. We consider the edge expansion of small sets, which is the probability of choosing a random vertex in the set and traversing a random edge touching it, and landing outside the set. A random small set of vertices has edge expansion nearly 1 with high probability. However, some sets in the Grassmann graph have strictly smaller edge expansion. We present a hypothesis that proposes a characterization of such sets: any such set must be denser inside subgraphs that are by themselves (isomorphic to) smaller Grassmann graphs. We say that such a set is non-pseudorandom. We achieve partial progress towards this hypothesis, proving that sets whose expansion is strictly smaller than 7/8 are non-pseudorandom. This is achieved through a spectral approach, showing that Boolean valued functions over the Grassmann graph that have significant correlation with eigenspaces corresponding to the top two non-trivial eigenvalues (that are approximately 1/2 and 1/4) must be non-pseudorandom.