Tight paths in fully directed hypergraphs

TL;DR

This work introduces and analyzes tight paths in fully directed hypergraphs, focusing on $r$-uniform $(r,k)$-tournaments and the existence of directed tight paths $P^{(r)}_s$. The authors connect threshold phenomena for path sizes to the pattern-shift graph $ ext{PSG}_r$, deriving exact growing-path thresholds via acyclic subgraph sizes and using Ramsey-type arguments, with precise results for small $r$ (notably $r\, ext{up to}\,5$). They establish a spectrum of regime results: constant, linear, and spanning-path thresholds, including a probabilistic spanning-path argument and density-based thresholds around density $1-rac{1}{r}$. The findings yield both concrete bounds and guiding conjectures, notably the $3$-uniform $(3,4)$-tournament conjecture, and provide a framework for extending these thresholds through PSG-based combinatorics. Overall, the paper advances understanding of when random-like fully directed hypergraphs force long or spanning tight paths, with implications for hypergraph orientations and Ramsey-type phenomena.

Abstract

It is well-known that every tournament has a spanning path. We consider hypergraph analogues. In an \emph{$r$-uniform fully directed hypergraph}, or \emph{$r$-digraph}, every edge is a list or $r$ distinct vertices. An $(r,k)$-tournament is an $r$-digraph $G$ such that for every $r$-set $S$ of vertices in $G$, exactly $k$ of the orderings of $S$ are edges in $G$. A \emph{directed tight path} is an $r$-digraph $G$ whose vertices can be ordered so that the intervals of size $r$ are the edges in $G$. Let $f(n,r,k)$ be the maximum $s$ such that every $n$-vertex $(r,k)$-tournament contains a tight path on $s$ vertices. Since every tournament has a spanning path, we have $f(n,2,1)=n$. In this paper, we show that the minimum $k$ such that $f(n,r,k)$ tends to infinity with $n$ is in the interval $\left[\left(1-\frac{1}{r}-O(\frac{\log r}{r^2\log\log r})\right)r!, ~\left(1-\frac{1}{r} - \frac{\varphi(r)-1}{r!}\right)r!\right]$ where $\varphi(r)$ is the Euler Totient Function, and we find the exact value when $r\le 5$. We also show that $Ω(\sqrt{\log n/\log \log n}) \le f(n,3,3) \le O(\log n)$ and $f(n,3,4)\ge Ω(n^{1/5})$.

Figures (3)

• Figure 1: $\mathrm{PSG}_3$
• Figure 2: The two shift cycles above (dotted edges) in $\mathrm{PSG}_4$ are replaced with four $2$-cycles (solid edges) in our construction of a family of disjoint cycles.
• Figure 3: Bounds on thresholds in $4$-uniform and $5$-uniform tournaments

Theorems & Definitions (70)

• Proposition 1
• proof
• Theorem 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Theorem 5
• proof