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.