Bounds on Linear Tur\'{a}n Number for Trees

TL;DR

This paper extends bounds on the maximum size of acyclic linear hypergraphs (trees) for higher uniformities, providing new constructions, exact bounds for certain hypertrees, and conjectures on others.

Contribution

It introduces new bounds and characterizations for linear Turán numbers of acyclic hypergraphs across various uniformities, expanding prior work on 3-uniform cases.

Findings

Lower bound for linear Turán number of r-uniform trees with k edges

Exact bound and characterization for hypertrees with four edges

Upper bound for crown hypergraphs and a conjecture for paths

Abstract

A hypergraph $H$ is said to be \emph{linear} if every pair of vertices lies in at most one hyperedge. Given a family $\mathcal{F}$ of $r$-uniform hypergraphs, an $r$-uniform hypergraph $H$ is \emph{$\mathcal{F}$-free} if it contains no member of $\mathcal{F}$ as a subhypergraph. The \emph{linear Tur\'{a}n number} $ex_r^{\mathrm{lin}}(n,\mathcal{F})$ denotes the maximum number of hyperedges in an $\mathcal{F}$-free linear $r$-uniform hypergraph on $n$ vertices. Gy\'arf\'as, Ruszink\'o, and S\'ark\"ozy~[\emph{Linear Tur\'an numbers of acyclic triple systems}, European J.\ Combin.\ (2022)] initiated the study of bounds on the linear Tur\'an number for acyclic $3$-uniform linear hypergraphs. In this paper, we extend the study of linear Tur\'{a}n numbers for acyclic systems to higher uniformity. We first give a construction for any linear $r$-uniform tree with $k$ edges that yields the lower bound $ ex_r^{\mathrm{lin}}(n,T_k^r)\ge {n(k-1)}/{r}, $ under mild divisibility and existence assumptions. Next, we study hypertrees with four edges. We prove the exact bound $ ex_r^{\mathrm{lin}}(n,B_4^r)\le {(r+1)n}/{r} $ and characterize the extremal hypergraph class, where $B_4^r$ is formed from $S_3^r$ by appending a hyperedge incident to a degree-one vertex. We also prove the bound $ ex_r^{\mathrm{lin}}(n,E_4^r)\le {(2r-1)n}/{r} $ for the crown $E_4^r$. Finally, we give a construction showing $ ex_r^{\mathrm{lin}}(n,P_4^r)\ge {(r+1)n}/{r} $ under suitable assumptions and conclude with a conjecture on sharp upper bound for $P_4^r$.