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 analogs of local and global LYM inequalities, the Erdős–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 Bollobás, 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, Patkós, Tompkins and Xiao on pairs of set systems satisfying both an intersection and a cross-intersection condition.

中文翻译：

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外代数中的组合数学和 Bollobás 两族定理

我们研究了有限维实向量空间的外代数的子空间的组合结构，与超图的极值组合并行工作。使用初始单项式、底层向量空间到子空间的投影以及内部积，我们找到了局部和全局 LYM 不等式的类比、Erdős-Ko-Rado 定理和 Ahlswede-Khachatrian 边界$吨$- 交叉超图。使用这些工具，我们证明了 Bollobás 的两族定理的新扩展，为满足倾斜交叉交叉条件的子空间配置提供了加权界限。我们还验证了 Gerbner、Keszegh、Methuku、Abhishek、Nagy、Patkós、Tompkins 和 Xiao 对满足交叉和交叉条件的集合系统的最近猜想。