Relative Tur\'an Numbers for Hypergraph Cycles

TL;DR

This paper establishes lower bounds on the maximum size of hypergraph subgraphs avoiding certain cycle configurations, specifically Berge cycles, in 3-uniform hypergraphs with bounded degree.

Contribution

It provides new lower bounds for relative Turán numbers for Berge cycles in 3-uniform hypergraphs, demonstrating bounds that are tight up to lower order terms.

Findings

Lower bounds for $ ext{ex}(H, ext{C}_4^{3})$ and $ ext{ex}(H, ext{C}_5^{3})$ in terms of maximum degree and total edges.

Bounds are tight up to the $o(1)$ term, indicating near-optimality.

Results extend understanding of cycle avoidance in hypergraph Turán problems.

Abstract

For an $r$-uniform hypergraph $H$ and a family of $r$-uniform hypergraphs $\mathcal{F}$, the relative Tur\'{a}n number $\mathrm{ex}(H,\mathcal{F})$ is the maximum number of edges in an $\mathcal{F}$-free subgraph of $H$. In this paper we give lower bounds on $\mathrm{ex}(H,\mathcal{F})$ for certain families of hypergraph cycles $\mathcal{F}$ such as Berge cycles and loose cycles. In particular, if $\mathcal{C}_\ell^3$ denotes the set of all $3$-uniform Berge $\ell$-cycles and $H$ is a 3-uniform hypergraph with maximum degree $\Delta$, we prove \[\mathrm{ex}(H,\mathcal{C}_4^{3})\ge \Delta^{-3/4-o(1)}e(H),\] \[\mathrm{ex}(H,\mathcal{C}_5^{3})\ge \Delta^{-3/4-o(1)}e(H),\] and these bounds are tight up to the $o(1)$ term.