### Abstract

A hypergraph is called an r×r grid if it is isomorphic to a pattern of r horizontal and r vertical lines, i.e.,a family of sets {A_{1}, ..., A_{r}, B_{1}, ..., B_{r}} such that A_{i}∩A_{j}=B_{i}∩B_{j}=φ for 1≤ii∩B_{j}{pipe}=1 for 1≤i, j≤r. Three sets C_{1}, C_{2}, C_{3} form a triangle if they pairwise intersect in three distinct singletons, {pipe}C_{1}∩C_{2}{pipe}={pipe}C_{2}∩C_{3}{pipe}={pipe}C_{3}∩C_{1}{pipe}=1, C_{1}∩C_{2}≠C_{1}∩C_{3}. A hypergraph is linear, if {pipe}E∩F{pipe}≤1 holds for every pair of edges E≠F.In this paper we construct large linear r-hypergraphs which contain no grids. Moreover, a similar construction gives large linear r-hypergraphs which contain neither grids nor triangles. For r≥. 4 our constructions are almost optimal. These investigations are motivated by coding theory: we get new bounds for optimal superimposed codes and designs.

Original language | English |
---|---|

Pages (from-to) | 302-324 |

Number of pages | 23 |

Journal | Advances in Mathematics |

Volume | 240 |

DOIs | |

Publication status | Published - Jun 2013 |

### Fingerprint

### Keywords

- Density problems
- Superimposed codes
- Turán hypergraph problem
- Union free hypergraphs

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

*240*, 302-324. https://doi.org/10.1016/j.aim.2013.03.009

**Uniform hypergraphs containing no grids.** / Füredi, Z.; Ruszinkó, Miklós.

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 240, pp. 302-324. https://doi.org/10.1016/j.aim.2013.03.009

}

TY - JOUR

T1 - Uniform hypergraphs containing no grids

AU - Füredi, Z.

AU - Ruszinkó, Miklós

PY - 2013/6

Y1 - 2013/6

N2 - A hypergraph is called an r×r grid if it is isomorphic to a pattern of r horizontal and r vertical lines, i.e.,a family of sets {A1, ..., Ar, B1, ..., Br} such that Ai∩Aj=Bi∩Bj=φ for 1≤ii∩Bj{pipe}=1 for 1≤i, j≤r. Three sets C1, C2, C3 form a triangle if they pairwise intersect in three distinct singletons, {pipe}C1∩C2{pipe}={pipe}C2∩C3{pipe}={pipe}C3∩C1{pipe}=1, C1∩C2≠C1∩C3. A hypergraph is linear, if {pipe}E∩F{pipe}≤1 holds for every pair of edges E≠F.In this paper we construct large linear r-hypergraphs which contain no grids. Moreover, a similar construction gives large linear r-hypergraphs which contain neither grids nor triangles. For r≥. 4 our constructions are almost optimal. These investigations are motivated by coding theory: we get new bounds for optimal superimposed codes and designs.

AB - A hypergraph is called an r×r grid if it is isomorphic to a pattern of r horizontal and r vertical lines, i.e.,a family of sets {A1, ..., Ar, B1, ..., Br} such that Ai∩Aj=Bi∩Bj=φ for 1≤ii∩Bj{pipe}=1 for 1≤i, j≤r. Three sets C1, C2, C3 form a triangle if they pairwise intersect in three distinct singletons, {pipe}C1∩C2{pipe}={pipe}C2∩C3{pipe}={pipe}C3∩C1{pipe}=1, C1∩C2≠C1∩C3. A hypergraph is linear, if {pipe}E∩F{pipe}≤1 holds for every pair of edges E≠F.In this paper we construct large linear r-hypergraphs which contain no grids. Moreover, a similar construction gives large linear r-hypergraphs which contain neither grids nor triangles. For r≥. 4 our constructions are almost optimal. These investigations are motivated by coding theory: we get new bounds for optimal superimposed codes and designs.

KW - Density problems

KW - Superimposed codes

KW - Turán hypergraph problem

KW - Union free hypergraphs

UR - http://www.scopus.com/inward/record.url?scp=84876301768&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84876301768&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2013.03.009

DO - 10.1016/j.aim.2013.03.009

M3 - Article

VL - 240

SP - 302

EP - 324

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -