论文标题
Turán和Ramsey的问题，用于交替多线性地图
Turán and Ramsey problems for alternating multilinear maps
论文作者
论文摘要
在超图和外部代数之间的连接指导下，我们研究了Turán和Ramsey型问题，用于交替进行多线性图。这项研究在于组合学，群体理论和代数几何形状的交集，起源于Lovász的作品（Proc。Sixt。IndernBritish Compinatorial Conf。，1977年），Buhler，Gupta和Harris（J. Algebra，1987），以及Feldman和Feldman和Propp（Adv。Math。，1992年）。 我们的主要结果是用于交替双线性图的Ramsey定理。给定$ s，t \ in \ mathbb {n} $，$ s，$ s，t \ geq 2 $，以及带有$ \ dim（v）= s \ cdot t^4 $的双线映射$ f：v \ times v \ to u $ to u $ dimension- $ t $ subpace $ w \ leq v $，这样$ \ dim（f（w，w））= \ binom {t} {2} $。该结果具有自然的群体理论（对于有限的$ p $ - 群体）和几何学（对于格拉曼尼亚人）的含义，并带来了针对各种各样的群体和格拉斯曼尼亚人的新拉姆齐型问题。
Guided by the connections between hypergraphs and exterior algebras, we study Turán 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 Lovász (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.
