Geodesic
Turán and Ramsey problems for alternating multilinear maps
Discrete analysis (2023) Cet article a éte moissonné depuis la source Scholastica

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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.
Publié le : 
@article{DAS_2023_a10,
     author = {Youming Qiao},
     title = {Tur\'an and {Ramsey} problems for alternating multilinear maps},
     journal = {Discrete analysis},
     year = {2023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DAS_2023_a10/}
}
TY  - JOUR
AU  - Youming Qiao
TI  - Turán and Ramsey problems for alternating multilinear maps
JO  - Discrete analysis
PY  - 2023
UR  - http://geodesic.mathdoc.fr/item/DAS_2023_a10/
LA  - en
ID  - DAS_2023_a10
ER  - 
%0 Journal Article
%A Youming Qiao
%T Turán and Ramsey problems for alternating multilinear maps
%J Discrete analysis
%D 2023
%U http://geodesic.mathdoc.fr/item/DAS_2023_a10/
%G en
%F DAS_2023_a10
Youming Qiao. Turán and Ramsey problems for alternating multilinear maps. Discrete analysis (2023). http://geodesic.mathdoc.fr/item/DAS_2023_a10/