Book contents
- Frontmatter
- Contents
- Preface
- Part I Extremal Graph Theory
- Part II Extremal Set Theory and Hypergraph Theory
- Introduction to Part II
- 11 Posets
- 12 Intersecting Families
- 13 Shadows and Matchings
- 14 Linear Algebra Method
- 15 Algebraic Methods
- 16 Sunflowers
- 17 The Turán Problem for Hypergraphs
- 18 Matchings in Linear Hypergraphs
- 19 Independent Sets in Hypergraphs
- 20 Hypergraph Regularity
- 21 Hypergraph Containers
- 22 Hypergraph Ramsey Theory
- Glossary of Notation
- References
- Index
14 - Linear Algebra Method
from Part II - Extremal Set Theory and Hypergraph Theory
Published online by Cambridge University Press: aN Invalid Date NaN
Book contents
- Frontmatter
- Contents
- Preface
- Part I Extremal Graph Theory
- Part II Extremal Set Theory and Hypergraph Theory
- Introduction to Part II
- 11 Posets
- 12 Intersecting Families
- 13 Shadows and Matchings
- 14 Linear Algebra Method
- 15 Algebraic Methods
- 16 Sunflowers
- 17 The Turán Problem for Hypergraphs
- 18 Matchings in Linear Hypergraphs
- 19 Independent Sets in Hypergraphs
- 20 Hypergraph Regularity
- 21 Hypergraph Containers
- 22 Hypergraph Ramsey Theory
- Glossary of Notation
- References
- Index
Summary
A versatile proof approach in extremal set theory is the Linear Algebra method. In order to show an upper bound on the number of sets satisfying certain conditions, we associate a vector to each set and show that the corresponding collection of vectors is linearly independent in an appropriate vector space. Then we use the elementary fact that the maximum size of a linearly independent set is at most the dimension of the space. This chapter explores this method in its myriad forms.
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- Extremal Graph and Hypergraph TheoryWith Ramsey Theory, pp. 243 - 260Publisher: Cambridge University PressPrint publication year: 2026
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