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Turán density of cliques of order five in $3$-uniform hypergraphs with quasirandom links

Sören Berger, Simón Piga, Christian Reiher, Vojtěch Rödl, Mathias Schacht

Abstract

We show that $3$-uniform hypergraphs with the property that all vertices have a quasirandom link graph with density bigger than $1/3$ contain a clique on five vertices. This result is asymptotically best possible.

Turán density of cliques of order five in $3$-uniform hypergraphs with quasirandom links

Abstract

We show that -uniform hypergraphs with the property that all vertices have a quasirandom link graph with density bigger than contain a clique on five vertices. This result is asymptotically best possible.
Paper Structure (18 sections, 21 theorems, 127 equations)

This paper contains 18 sections, 21 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Reduced hypergraphs and bicolourings
  3. Preliminaries
  4. Transversals and cherries
  5. Inhabited transversals in weakly dense reduced hypergraphs
  6. Partite versions
  7. Bicolouring wicked reduced hypergraphs
  8. Plan
  9. Naïve ideas
  10. A more realistic picture
  11. Organisation
  12. Holes and links in reduced hypergraphs
  13. Intersecting and disjoint links
  14. Equivalent holes
  15. Unions of equivalent holes
  16. ...and 3 more sections

Key Result

Theorem 1.2

For every integer $t\geq 2$ we have

Theorems & Definitions (53)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3: Main result
  • Example 1.4
  • Corollary 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • ...and 43 more