Entropy contraction for Kneser graphs and optimality of product spaces as inclusion samplers

Establish optimal entropy contraction for Kneser graphs (the down-up walk on the swap complex), which would imply optimal top-level inclusion sampling for partite high dimensional expanders. Additionally, determine whether product spaces are optimal inclusion samplers among general complexes.

Background

The proof of inclusion sampling for partite complexes requires moving to a lower-dimensional skeleton; optimal sampling at the top level is not proved—even for product spaces. The appendix notes that such optimality would follow from proving optimal entropy contraction for Kneser graphs.

Resolving this would close the remaining gap between the independent sampling benchmark and partite HDX, and clarify whether product spaces (complete complexes) are extremal inclusion samplers.