The classification of Boolean degree $1$ functions in high-dimensional finite vector spaces

We classify the {\it Boolean degree $1$ functions} of $k$-spaces in a vector space of dimension $n$ (also known as {\it Cameron-Liebler classes}) over the field with $q$ elements for $n \geq n_0(k, q)$, a problem going back to a work by Cameron and Liebler from 1982. This also implies that two-intersecting sets with respect to $k$-spaces do not exist for $n \geq n_0(k, q)$. Our main ingredient is the Ramsey theory for geometric lattices.