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Small set expansion in Johnson graphs (=slices of hypercube) Moshkovitz, Danna


For natural numbers t<l<k, the nodes of the Johnson 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 Johnson 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 necessarily small set expanders like the noisy hypercube -- can only have small non-expanding sets of a certain simple structure. The result was motivated, in part, by works of Dinur, Kindler, Khot, Minzer and Safra, which hypothesized and made partial progress on a similar result for the Grassmann graph, and showed that it would imply the proof of the 2 to 2 Conjecture in PCP. In turn, our result served as a crucial step towards the full result for the Grassmann graphs completed subsequently by Khot, Minzer and Safra. This is joint work with Subhash Khot, Dor Minzer and Muli Safra

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