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

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