On growth functions of ordered hypergraphs

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.