Optimal inclusion sampling at the top level via entropy contraction for Kneser graphs

Establish optimal inclusion sampling at the top level for partite high dimensional expanders and, in particular, for product spaces by proving optimal entropy contraction for Kneser graphs. Concretely, prove the requisite entropy contraction bound that would imply top-level Chernoff-type inclusion sampling, or determine whether product spaces are optimal inclusion samplers.

Background

The paper’s inclusion sampling proof for partite HDX currently operates on a slightly lower dimensional skeleton; the authors cannot prove optimal top-level inclusion sampling even for product spaces.

They show in the appendix that optimal entropy contraction for Kneser graphs would imply the desired top-level bounds, suggesting a concrete analytic target whose resolution would settle the sampling question.

References

In fact, we are unable to prove optimal inclusion sampling at the top level even for product spaces. In \pref{app:swap-complex}, we show this (and the general bound for HDX) would be implied by proving optimal entropy contraction of the Kneser graphs. Does such a bound hold? Are product spaces optimal inclusion samplers?

— Chernoff Bounds and Reverse Hypercontractivity on HDX  (2404.10961 - Dikstein et al., 2024) in Open questions, Item 3