Paths in hypergraphs: a rescaling phenomenon

TL;DR

This paper investigates the minimum maximum degree in hypergraphs avoiding certain paths, revealing a rescaling phenomenon where the degree drops sharply at specific edge densities.

Contribution

It characterizes the structure of extremal hypergraphs and demonstrates a phase transition in degree behavior for particular path lengths.

Findings

Degree drops from Θ(n^2) to Θ(n) at m ~ n^2/8

Provides structural characterization of extremal hypergraphs

Identifies a rescaling phenomenon in hypergraph degrees

Abstract

Let $P^k_\ell$ denote the loose $k$-path of length $\ell$ and let define $f^k_\ell(n,m)$ as the minimum value of $\Delta(H)$ over all $P^k_\ell$-free $k$-graphs $H$ with $n$ vertices and $m$ edges. In the paper we study the behavior of $f^4_2(n,m)$ and $f^3_3(n,m)$ and characterize the structure of extremal hypergraphs. In particular, it is shown that when $m\sim n^2/8$ the value of each of these functions drops down from $\Theta(n^2)$ to $\Theta(n)$.