$n$ vertices with $R(G)\subseteq R$. We simplify it to $G$ if $n$ and $R$ are clear under context.

Let $R$ be a fixed set of edge types. Let $H$ be an $R$-graph. The number of vertices in $H$ is denoted by $v(H):=|V(H)|$. Our goal is to measure the edge density of $H$ and be able to compare it (in a meaningful way) to the edge density of other $R$ graphs. The standard way to measure this density would be: $$\mu(H) = \frac{|E(H)|}{\sum_{k\in R(H)}\binom{v(H)}{k}}.$$ This density ranges from 0 to 1 (as one would expect)–a complete $R$-graph having a density of 1. Unfortunately, this is no longer a useful measure of density since the number of edges with maximum cardinality will dwarf the number of edges of all other sizes. Specifically, one could take $k$-uniform hypergraph (where $k=\max\{r:r\in R(H)\}$) on enough vertices and make its density as close to 1 he likes. The problem is that this $k$-uniform hypergraph is quite different from the complete $R$-graph (when $|R|>1$) with the same number of vertices. Instead, we use the Lubell function to measure the edge density. This is adapted from the use of the Lubell function studying families of subsets.

For a 