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Extremal Numbers of Hypergraph Suspensions of Even Cycles

Sayan Mukherjee

Abstract

For fixed $k\ge 2$, determining the order of magnitude of the number of edges in an $n$-vertex bipartite graph not containing $C_{2k}$, the cycle of length $2k$, is a long-standing open problem. We consider an extension of this problem to triple systems. In particular, we prove that the maximum number of triples in an $n$-vertex triple system which does not contain a $C_6$ in the link of any vertex, has order of magnitude $n^{7/3}$. Additionally, we construct new families of dense $C_6$-free bipartite graphs with $n$ vertices and $n^{4/3}$ edges in order of magnitude.

Extremal Numbers of Hypergraph Suspensions of Even Cycles

Abstract

For fixed , determining the order of magnitude of the number of edges in an -vertex bipartite graph not containing , the cycle of length , is a long-standing open problem. We consider an extension of this problem to triple systems. In particular, we prove that the maximum number of triples in an -vertex triple system which does not contain a in the link of any vertex, has order of magnitude . Additionally, we construct new families of dense -free bipartite graphs with vertices and edges in order of magnitude.

Paper Structure

This paper contains 12 sections, 10 theorems, 44 equations, 2 tables.

Table of Contents

  1. Introduction
  2. 3-Graphs
  3. Graphs
  4. Lower bounds on $\mathrm{ex}_3(n,\widetilde{C}_{2k})$
  5. The 3-graphs $\mathrm{D}_3(k,q)$
  6. Proof of Proposition \ref{['prop:a-to-a1-000']}
  7. Proof of Proposition \ref{['prop:D3kq-twolinks']}
  8. The bipartite graphs $\mathrm{D}'(k,q)$
  9. Proof of Theorem \ref{['thm:c2ktilde']}
  10. The arc construction and Wenger's construction
  11. Proof of Proposition \ref{['prop:Arc_Wenger_isomorphism']}
  12. Proof of Theorem \ref{['thm:newC6construction']}

Key Result

Proposition 1.1

For $k\ge 2$,

Theorems & Definitions (23)

  • Proposition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark
  • Proposition 1.4
  • Theorem 1.5
  • proof : Proof of Proposition \ref{['prop:probabilistic-lower-C2k']}
  • Definition 2.1: The bipartite graphs $\mathrm{D}(q)$
  • Proposition 2.2
  • Definition 2.3: The 3-partite 3-graphs $\mathrm{D}_3(q)$
  • ...and 13 more