Bounds on Linear Turán Number for Trees

TL;DR

This work advances the study of linear Turán numbers for higher-uniformity trees by leveraging graph expansions and Steiner-system designs to derive tight density bounds and structural characterizations. The authors provide a general construction yielding $ex_r^{\mathrm{lin}}(n,T_k^r) \ge {n(k-1)}/{r}$ and prove sharp bounds for several four-edge hypertrees, notably $B_4^r$, with extremal configurations tied to unions of Steiner systems $S(2,r,r^2)$. They also establish an upper bound for the crown $E_4^r$ and a lower bound for the four-edge path $P_4^r$, culminating in a conjecture that the bound for $P_4^r$ is tight and realized by $S(2,r,r^2)$-based constructions. Overall, the paper provides exact extremal results under Steiner-system existence assumptions and guides sharpness questions for broader classes of linear hypertrees.

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 Turán 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. Gyárfás, Ruszinkó, and Sárközy~[\emph{Linear Turán numbers of acyclic triple systems}, European J.\ Combin.\ (2022)] initiated the study of bounds on the linear Turán number for acyclic $3$-uniform linear hypergraphs. In this paper, we extend the study of linear Turán number 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 $P_4^r$.

Figures (1)

• Figure 1: Configurations of $B_4^r$ and $E_4^r$ (in the figure $r =5$).

Theorems & Definitions (38)

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