Tur\'an Numbers of Ordered Tight Hyperpaths

TL;DR

This paper determines the exact Turán numbers for certain ordered hyperpaths in hypergraphs when specific parameters are met, and provides conjectures and constructions for cases still unresolved.

Contribution

It establishes exact Turán numbers for ordered hyperpaths in hypergraphs for specific parameter ranges and introduces new constructions and conjectures for open cases.

Findings

Exact Turán numbers for $r$-uniform $s$-vertex tight paths when $r extless s extless 2r$ and $n$ even.

Asymptotic formula for the Turán number when $r extless s extless 2r$.

A new construction for $r=3$ hypergraphs that may be extremal for avoiding certain hyperpaths.

Abstract

An ordered hypergraph is a hypergraph $G$ whose vertex set $V(G)$ is linearly ordered. We find the Tur\'an numbers for the $r$-uniform $s$-vertex tight path $P^{(r)}_s$ (with vertices in the natural order) exactly when $r\le s < 2r$ and $n$ is even; our results imply $\mathrm{ex}_{>}(n,P^{(r)}_s)=(1-\frac{1}{2^{s-r}} + o(1))\binom{n}{r}$ when $r\le s<2r$. When $r\ge 2s$, the asymptotics of $\mathrm{ex}_{>}(n,P^{(r)}_s)$ remain open. For $r=3$, we give a construction of an $r$-uniform $n$-vertex hypergraph not containing $P^{(r)}_s$ which we conjecture to be asymptotically extremal.