Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Turán number of Berge paths

Xin Cheng, Dániel Gerbner, Hilal Hama Karim, Shujing Miao, Junpeng Zhou

Abstract

A Berge path of length $k$ in an $r$-uniform hypergraph is a collection of $k$ hyperedges $h_1,\dots,h_k$ and $k+1$ vertices $v_1,\dots,v_{k+1}$ such that $v_i, v_{i+1}\in h_i$ for each $1\le i\le k$. Győri, Katona and Lemons [\textit{European J. Combin. 58 (2016) 238--246}] generalized the Erdős-Gallai theorem to Berge paths and established bounds for the Turán number of Berge paths. However, these bounds are sharp only when some divisibility conditions hold. Gy\H ori, Lemons, Salia and Zamora [\textit{J. Combin. Theory Ser. B 148 (2021) 239--250}] determined the exact value of the Turán number of Berge paths in the case $k\le r$. In this paper, we settle the final open case $k>r$, thereby completing the determination of the Turán number of Berge paths.

The Turán number of Berge paths

Abstract

A Berge path of length in an -uniform hypergraph is a collection of hyperedges and vertices such that for each . Győri, Katona and Lemons [\textit{European J. Combin. 58 (2016) 238--246}] generalized the Erdős-Gallai theorem to Berge paths and established bounds for the Turán number of Berge paths. However, these bounds are sharp only when some divisibility conditions hold. Gy\H ori, Lemons, Salia and Zamora [\textit{J. Combin. Theory Ser. B 148 (2021) 239--250}] determined the exact value of the Turán number of Berge paths in the case . In this paper, we settle the final open case , thereby completing the determination of the Turán number of Berge paths.
Paper Structure (3 sections, 12 theorems, 7 equations)

This paper contains 3 sections, 12 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main']} for $k=r+1$
  3. Proof of Theorem \ref{['main']} for $k>r+1$

Key Result

Theorem 1.2

If $k\ge r+1>3$, then ${\rm{ex}}_r(n,{\rm Berge}{\text{-}}P_k)\leq \frac{n}{k}\binom{k}{r}$. Furthermore, this bound is sharp whenever $k$ divides $n$. If $r\geq k>2$, then ${\rm{ex}}_r(n,{\rm Berge}{\text{-}}P_k)\leq \frac{n(k-1)}{r+1}$. Furthermore, this bound is sharp whenever $r+1$ divides $n$.

Theorems & Definitions (29)

  • Definition 1.1: Győri, Katona and Lemons B6
  • Theorem 1.2: Győri, Katona, Lemons B6, Davoodi, Győri, Methuku, Tompkins 10-10A1
  • Theorem 1.3
  • Theorem 1.4: Chakraborti and Chen chch
  • Lemma 2.1
  • proof : Proof
  • Lemma 2.2
  • proof : Proof
  • Lemma 2.3
  • proof : Proof
  • ...and 19 more