Here's a neat result I heard about at virtual Dagstuhl last week, a new lower bound on the number of hyperplanes that cuts all the edges of a hypercube.
A n-dimensional hypercube has 2n vertices corresponding to the binary strings of length n. Edges are between two vertices that differ in exactly one bit, for a total of (n/2) 2n edges. Hypercubes played a large role in Hao Huang's recent proof of the sensitivity conjecture.
A hyperplane is just a linear inequality in x1,…,xn the bits of the string corresponding to a vertex. An edge is cut if the inequality is true for one vertex and false for the other.
With n hyperplanes you can cut all the edges in two very different ways.
- The hyperplanes xi > 0 for each i. These are n orthogonal hyperplanes.
- The hyperplanes x1+…+xn > i for each i from 0 to n-1. These are n parallel hyperplanes.