# Combinatorics in the exterior algebra and the Bollob\'{a}s Two Families Theorem

## TL;DR

This paper explores the combinatorial properties of subspaces in the exterior algebra, establishing analogues of classical combinatorial theorems and extending the Bollobás Two Families Theorem with new bounds and verified conjectures.

## Contribution

It introduces novel algebraic tools to extend classical extremal combinatorics results to subspace configurations, including a weighted extension of the Two Families Theorem.

## Key findings

- Established algebraic analogues of LYM inequalities and Erdős–Ko–Rado theorem.
- Proved a weighted extension of the Two Families Theorem for subspace configurations.
- Verified a recent conjecture on pairs of set systems with intersection properties.

## Abstract

We investigate the combinatorial structure of subspaces of the exterior algebra of a finite-dimensional real vector space, working in parallel with the extremal combinatorics of hypergraphs. Using initial monomials, projections of the underlying vector space onto subspaces, and the interior product, we find analogues of local and global LYM inequalities, the Erd\H{o}s-Ko-Rado theorem, and the Ahlswede-Khachatrian bound for $t$-intersecting hypergraphs. Using these tools, we prove a new extension of the Two Families Theorem of Bollob\'{a}s, giving a weighted bound for subspace configurations satisfying a skew cross-intersection condition. We also verify a recent conjecture of Gerbner, Keszegh, Methuku, Abhishek, Nagy, Patk\'{o}s, Tompkins, and Xiao on pairs of set systems satisfying both an intersection and a cross-intersection condition.