Turán and Ramsey Problems for Alternating Multilinear Maps

Qiao, Y

Turán and Ramsey Problems for Alternating Multilinear Maps

Qiao, Y

Permalink

Publication Type:: Journal Article
Citation:: Discrete Analysis, 2023, 2023
Issue Date:: 2023-01-01

Open Access

Copyright Clearance Process

Recently Added
In Progress
Open Access

This item is open access.

Adobe PDF

Download Published versionAdobe PDF (316.93 kB)

View on publisher's site

View statistics

Full metadata record

Field	Value	Language
dc.contributor.author	Qiao, Y https://orcid.org/0000-0003-4334-1449
dc.date.accessioned	2024-03-06T04:18:26Z
dc.date.available	2024-03-06T04:18:26Z
dc.date.issued	2023-01-01
dc.identifier.citation	Discrete Analysis, 2023, 2023
dc.identifier.issn	2397-3129
dc.identifier.uri	http://hdl.handle.net/10453/176196
dc.description.abstract	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 ∈ N, s, t ≥ 2, and an alternating bilinear map ϕ: V × V → U with dim(V) ≥ s ·t4, we show that there exists either a dimension-s subspaceW ≤ V such that dim(span(ϕ(W,W))) = 0, or a dimension-t subspaceW ≤ V such that dim(span(ϕ(W,W))) = (t2). 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.
dc.language	en
dc.relation	http://purl.org/au-research/grants/arc/DP200100950
dc.relation.ispartof	Discrete Analysis
dc.relation.isbasedon	10.19086/da.84736
dc.rights	info:eu-repo/semantics/openAccess
dc.title	Turán and Ramsey Problems for Alternating Multilinear Maps
dc.type	Journal Article
utslib.citation.volume	2023
pubs.organisational-group	University of Technology Sydney
pubs.organisational-group	University of Technology Sydney/Faculty of Engineering and Information Technology
pubs.organisational-group	University of Technology Sydney/Strength - QSI - Centre for Quantum Software and Information
pubs.organisational-group	University of Technology Sydney/Faculty of Engineering and Information Technology/School of Computer Science
utslib.copyright.status	open_access	*
dc.date.updated	2024-03-06T04:18:25Z
pubs.publication-status	Published
pubs.volume	2023

Abstract:

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 ∈ N, s, t ≥ 2, and an alternating bilinear map ϕ: V × V → U with dim(V) ≥ s ·t4, we show that there exists either a dimension-s subspaceW ≤ V such that dim(span(ϕ(W,W))) = 0, or a dimension-t subspaceW ≤ V such that dim(span(ϕ(W,W))) = (t2). 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.

Please use this identifier to cite or link to this item:

http://hdl.handle.net/10453/176196