Extremal subsets of the boolean cube with respect to projections

In this talk, we will present an asymptotically tight bound on the largest number of distinct projections onto \alpha n coordinates guaranteed in any family of n^r binary vectors of length n, where 0 < \alpha \le 1 and r = n^{o(1)}, thus closing a gap open since a work of Bollobas and Radcliffe from 1995. For the proof, we establish a “sparse” version of a classical result, the Kruskal-Katona Theorem, where we give a stronger guarantee when the hypergraph does not induce dense subhypergraphs. We also present a geometric application regarding point-halfspace separation in R^d.

The talk is based on a recent joint work with Noga Alon and Guy Moshkovitz.