# Turán Numbers of Complete 3-Uniform Berge-Hypergraphs

Turán Numbers of Complete 3-Uniform Berge-Hypergraphs Given a family $${\mathcal {F}}$$ F of r-graphs, the Turán number of $${\mathcal {F}}$$ F for a given positive integer N, denoted by $$ex(N,{\mathcal {F}})$$ e x ( N , F ) , is the maximum number of edges of an r-graph on N vertices that does not contain any member of $${\mathcal {F}}$$ F as a subgraph. For given $$r\ge 3$$ r ≥ 3 , a complete r-uniform Berge-hypergraph, denoted by $${K}_n^{(r)}$$ K n ( r ) , is an r-uniform hypergraph of order n with the core sequence $$v_{1}, v_{2}, \ldots ,v_{n}$$ v 1 , v 2 , … , v n as the vertices and distinct edges $$e_{ij},$$ e ij , $$1\le i<j\le n,$$ 1 ≤ i < j ≤ n , where every $$e_{ij}$$ e ij contains both $$v_{i}$$ v i and $$v_{j}$$ v j . Let $${\mathcal {F}}^{(r)}_n$$ F n ( r ) be the family of complete r-uniform Berge-hypergraphs of order n. We determine precisely $$ex(N,{\mathcal {F}}^{(3)}_{n})$$ e x ( N , F n ( 3 ) ) for $$N \ge n \ge 13$$ N ≥ n ≥ 13 . We also find the extremal hypergraphs avoiding $${\mathcal {F}}^{(3)}_{n}$$ F n ( 3 ) . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Graphs and Combinatorics Springer Journals

# Turán Numbers of Complete 3-Uniform Berge-Hypergraphs

, Volume 34 (4) – May 28, 2018
14 pages
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Publisher
Springer Japan
Copyright
Copyright © 2018 by Springer Japan KK, part of Springer Nature
Subject
Mathematics; Combinatorics; Engineering Design
ISSN
0911-0119
eISSN
1435-5914
D.O.I.
10.1007/s00373-018-1900-1
Publisher site
See Article on Publisher Site

### Abstract

Given a family $${\mathcal {F}}$$ F of r-graphs, the Turán number of $${\mathcal {F}}$$ F for a given positive integer N, denoted by $$ex(N,{\mathcal {F}})$$ e x ( N , F ) , is the maximum number of edges of an r-graph on N vertices that does not contain any member of $${\mathcal {F}}$$ F as a subgraph. For given $$r\ge 3$$ r ≥ 3 , a complete r-uniform Berge-hypergraph, denoted by $${K}_n^{(r)}$$ K n ( r ) , is an r-uniform hypergraph of order n with the core sequence $$v_{1}, v_{2}, \ldots ,v_{n}$$ v 1 , v 2 , … , v n as the vertices and distinct edges $$e_{ij},$$ e ij , $$1\le i<j\le n,$$ 1 ≤ i < j ≤ n , where every $$e_{ij}$$ e ij contains both $$v_{i}$$ v i and $$v_{j}$$ v j . Let $${\mathcal {F}}^{(r)}_n$$ F n ( r ) be the family of complete r-uniform Berge-hypergraphs of order n. We determine precisely $$ex(N,{\mathcal {F}}^{(3)}_{n})$$ e x ( N , F n ( 3 ) ) for $$N \ge n \ge 13$$ N ≥ n ≥ 13 . We also find the extremal hypergraphs avoiding $${\mathcal {F}}^{(3)}_{n}$$ F n ( 3 ) .

### Journal

Graphs and CombinatoricsSpringer Journals

Published: May 28, 2018

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