On growth functions of ordered hypergraphs

TL;DR

This paper investigates the growth rates of ideals of ordered hypergraphs with colored edges, establishing dichotomies that classify their growth as either constant, linear, polynomial, or at least a specific recursive sequence.

Contribution

It extends known results from graphs to hypergraphs, providing new dichotomies for growth functions of ideals of ordered hypergraphs with edge colorings.

Findings

Growth functions are either eventually constant or at least linear for all hypergraph ideals.

For 3-uniform hypergraphs with two colors, growth is either polynomial or at least as large as a specific recursive sequence.

Lower bounds in the dichotomies are proven to be tight.

Abstract

For $k,l\ge2$ we consider ideals of edge $l$-colored complete $k$-uniform hypergraphs $(n,\chi)$ with vertex sets $[n]=\{1, 2, \dots n\}$ for $n\in\mathbb{N}$. An ideal is a set of such colored hypergraphs that is closed to the relation of induced ordered subhypergraph. We obtain analogues of two results of Klazar [arXiv:0703047] who considered graphs, namely we prove two dichotomies for growth functions of such ideals of colored hypergraphs. The first dichotomy is for any $k,l\ge2$ and says that the growth function is either eventually constant or at least $n-k+2$. The second dichotomy is only for $k=3,l=2$ and says that the growth function of an ideal of edge two-colored complete $3$-uniform hypergraphs grows either at most polynomially, or for $n\ge23$ at least as $G_n$ where $G_n$ is the sequence defined by $G_1=G_2=1$, $G_3=2$ and $G_n = G_{n-1} + G_{n-3}$ for $n\ge4$. The lower bounds in both dichotomies are tight.