  Guided by the connections between hypergraphs and exterior algebras, we study
Tur\'an and Ramsey type problems for alternating multilinear maps. This study
lies at the intersection of combinatorics, group theory, and algebraic
geometry, and has origins in the works of Lov\'asz (Proc. Sixth British
Combinatorial Conf., 1977), Buhler, Gupta, and Harris (J. Algebra, 1987), and
Feldman and Propp (Adv. Math., 1992).
  Our main result is a Ramsey theorem for alternating bilinear maps. Given $s,
t\in \mathbb{N}$, $s, t\geq 2$, and an alternating bilinear map $f:V\times V\to
U$ with $\dim(V)=s\cdot t^4$, we show that there exists either a dimension-$s$
subspace $W\leq V$ such that $\dim(f(W, W))=0$, or a dimension-$t$ subspace
$W\leq V$ such that $\dim(f(W, W))=\binom{t}{2}$. This result has natural
group-theoretic (for finite $p$-groups) and geometric (for Grassmannians)
implications, and leads to new Ramsey-type questions for varieties of groups
and Grassmannians.
