Tur\'an and Ramsey problems for alternating multilinear maps

TL;DR

This paper explores Turán and Ramsey problems for alternating multilinear maps, establishing a Ramsey theorem for bilinear maps that links combinatorics, algebra, and geometry, with implications for group theory and Grassmannians.

Contribution

The paper proves a new Ramsey theorem for alternating bilinear maps, connecting combinatorial properties with algebraic and geometric structures, and introduces related open questions.

Findings

Established a Ramsey theorem for alternating bilinear maps.

Identified subspace structures with specific map dimensions.

Linked results to group theory and geometric contexts.

Abstract

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.