This paper studies expansion properties of the (generalized) Johnson Graph. For natural numbers
t < l < k, the nodes of the graph are sets of size l in a universe of size k. Two sets are connected if
their intersection is of size t. The Johnson graph arises often in combinatorics and theoretical computer
science: it represents a “slice” of the noisy hypercube, and it is the graph that underlies direct product
tests as well as a candidate hard unique game.
We prove that any small set of vertices in the graph either has near perfect edge expansion or is
not pseudorandom. Here “not pseudorandom” means that the set becomes denser when conditioning
on containing a small set of elements. In other words, we show that slices of the noisy hypercube –
while not small set expanders like the noisy hypercube – only have non-expanding small sets of a certain
simple structure.
This paper is related to a recent line of work establishing the 2-to-2 Theorem in PCP. The result
was motivated, in part, by [DKKMS] which hypothesized and made partial progress on similar result for the
Grassmann graph. In turn, our result for the Johnson graphs served as a crucial step towards the full
result for the Grassmann graphs completed subsequently in [KMS].