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The feasible region of hypergraphs.
- Source :
-
Journal of Combinatorial Theory - Series B . May2021, Vol. 148, p23-59. 37p. - Publication Year :
- 2021
-
Abstract
- Let F be a family of r -uniform hypergraphs. The feasible region Ω (F) of F is the set of points (x , y) in the unit square such that there exists a sequence of F -free r -uniform hypergraphs whose shadow density approaches x and whose edge density approaches y. The feasible region provides a lot of combinatorial information, for example, the supremum of y over all (x , y) ∈ Ω (F) is the Turán density π (F) , and Ω (∅) gives the Kruskal-Katona theorem. We undertake a systematic study of Ω (F) , and prove that Ω (F) is completely determined by a left-continuous almost everywhere differentiable function; and moreover, there exists an F for which this function is not continuous. We also extend some old related theorems. For example, we generalize a result of Fisher and Ryan to hypergraphs and extend a classical result of Bollobás by almost completely determining the feasible region for cancellative triple systems. [ABSTRACT FROM AUTHOR]
- Subjects :
- *HYPERGRAPHS
*DIFFERENTIABLE functions
*POINT set theory
Subjects
Details
- Language :
- English
- ISSN :
- 00958956
- Volume :
- 148
- Database :
- Academic Search Index
- Journal :
- Journal of Combinatorial Theory - Series B
- Publication Type :
- Academic Journal
- Accession number :
- 148988333
- Full Text :
- https://doi.org/10.1016/j.jctb.2020.12.004