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A Jump in the Codegree Turán Densities of Long Tight Cycles

József Balogh, Haoran Luo, Maya Sankar

Abstract

We study the codegree Turán density of $\mathcal{C}_\ell^r$, the $r$-uniform hypergraph tight cycle of length $\ell$. A result of Han, Lo, and Sanhueza-Matamala states that if $\ell$ is sufficiently large and $r/\gcd(r,\ell)$ is even, then the codegree Turán density of $\mathcal{C}_\ell^r$ is $1/2$. We prove that whenever the latter assumption is not satisfied, there is a significant drop in the codegree Turán density. That is, if $\ell$ is sufficiently large and $r/\gcd(r,\ell)$ is odd, then the codegree Turán density of $\mathcal{C}_\ell^r$ can be at most $1/3$. Moreover, this bound is tight for infinitely many uniformities $r$ and all sufficiently large $\ell$ in the corresponding residue classes modulo $r$. Our proof makes use of a group-theoretic connection between Turán-type theorems for tight cycles and ``oriented colorings'' of the edge set of a hypergraph.

A Jump in the Codegree Turán Densities of Long Tight Cycles

Abstract

We study the codegree Turán density of , the -uniform hypergraph tight cycle of length . A result of Han, Lo, and Sanhueza-Matamala states that if is sufficiently large and is even, then the codegree Turán density of is . We prove that whenever the latter assumption is not satisfied, there is a significant drop in the codegree Turán density. That is, if is sufficiently large and is odd, then the codegree Turán density of can be at most . Moreover, this bound is tight for infinitely many uniformities and all sufficiently large in the corresponding residue classes modulo . Our proof makes use of a group-theoretic connection between Turán-type theorems for tight cycles and ``oriented colorings'' of the edge set of a hypergraph.
Paper Structure (9 sections, 9 theorems, 18 equations, 4 figures)

This paper contains 9 sections, 9 theorems, 18 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation.
  3. \ref{['construction']} is $\mathcal{C}^r_{\ell}$-free
  4. $\mathcal{C}_{\equiv k}^r$-hom-free hypergraphs
  5. Permutations and oriented colorings
  6. Accordant colorings by $S_i\times S_{r-i}$
  7. Restricting to colors of the form $S_i\times S_{r-i}$
  8. Proof of the main theorem
  9. Concluding remarks

Key Result

Theorem 1

For every uniformity $r \geqslant 2$ and residue class $k \not\equiv 0 \pmod r$ such that $r / \gcd(r,k)$ is odd, there is an integer $L$ such that for every $\ell > L$ with $\ell \equiv k \pmod r$, we have

Figures (4)

  • Figure 1: Two edges $e_1,e_2$ colored with $S_4\times S_3$ in a $7$-graph. The vertices $+$ and $-$ are the vertices in $h_i^+$ and $h_i^-$, respectively. The shared $6$-tuple is of the same coloring in these two edges. Note that the signs $\pm$ represent colorings within $e_1$ and $e_2$; if there were an edge $e_3$ containing the rightmost five vertices and two extra vertices, then $e_3$ is not necessarily even colored with $S_4 \times S_3$.
  • Figure 2: $r = 6$ and $i = 3$. The coloring of every vertex $v \in X^+ \setminus e$ in $U+v$ is the same as $a_3$ in $e$. The coloring of every vertex $v \in X^- \setminus e$ in $U+v$ is the same as $b_3$ in $e$.
  • Figure 3: $r=5$ and $i=2$. Both $U+x_1+v$ and $U+x_2+v$ are edges in $\mathcal{H}$. Hence, the coloring of $x_1$ in $U+x_1$ should be the same as the coloring of $x_2$ in $U+x_2$.
  • Figure 4: $r = 8$ and $i = 3$. Let $e$ be an edge colored with $S_3 \times S_5$ and $U$ be a $3$-good set in $e$. By \ref{['colorInBetween']}, we have that all the $(3,4)$-sets of $(X^+(U), X^-(U))$ are of $S_3 \times S_5$, in which vertices in $X^+(U)$ are of $S_3$ and vertices in $X^-(U)$ are of $S_5$. Hence, by the coloring, no $(4,4)$-set of $(X^+(U),X^-(U))$ can be an edge.

Theorems & Definitions (35)

  • Theorem 1
  • Remark 3
  • Lemma 4
  • proof
  • Definition 5
  • Definition 6
  • Theorem 7: Theorem 3.4 in sankar2024turan
  • Example 8
  • Lemma 9
  • proof
  • ...and 25 more