Uniform hypergraphs containing no grids

Z. Füredi, Miklós Ruszinkó

Research output: Contribution to journalArticle

3 Citations (Scopus)

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 {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.

Original languageEnglish
Pages (from-to)302-324
Number of pages23
JournalAdvances in Mathematics
Volume240
DOIs
Publication statusPublished - Jun 2013

Fingerprint

Uniform Hypergraph
Hypergraph
Grid
Triangle
Optimal Codes
Coding Theory
Intersect
Pairwise
Horizontal
Isomorphic
Vertical
Distinct
Line

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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

In: Advances in Mathematics, Vol. 240, 06.2013, p. 302-324.

Research output: Contribution to journalArticle

Füredi, Z. ; Ruszinkó, Miklós. / Uniform hypergraphs containing no grids. In: Advances in Mathematics. 2013 ; Vol. 240. pp. 302-324.
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