The paper investigates expansion properties of the Grassmann graph, motivated by recent results of [KMS, DKKMS] 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.