a random $n$-permutation $\sigma$ uniformly. Let $X$ be the number of edges of $G^R_n$ hit by a random flag $\sigma(L)$. Note that each edge $F$ has probability $\frac{1}{{n\choose |F|}}$ of being hit by $\sigma(L)$. We have $$\label{eq:ef}
    {{\rm E}}(X)=\sum_{F\in E(G)}\frac{1}{{n\choose |F|}}=h_n(G).$$ Since $G^R_n$ is $L$-free, we have $X\leq r-1$. Taking the expectation, we have $$h_n(G^R_n)={{\rm E}}(X)\leq r-1.$$ Hence, $\pi_n(H)\leq r-1$. The result is followed after combining with item 1. $\square$

A hypergraph $H$ is called [*degenerate*]{} if $\pi(H)=|R(H)|-1$.

By Proposition \[p1\], flags, and specifically chains, are degenerate hypergraphs. A necessary condition for $H$ to be degenerate is that every level hypergraph $H^{k_i}$ is $k_i$-partite. The following examples will show that the converse is not true.

[**Example 1:**]{} The complete hypergraph $K^{\{1,2\}}_2$ has three edges $\{1\}$, $\{2\}$, and $\{1,2\}$. We claim $$\label{eq:k12}
   \pi(K^{\{1,2\}}_2)=\frac{5}{4}.$$ The lower bound is from the following construction. Partition $[n]$ into two parts $A$ and $B$ of nearly equal size. Consider the hypergraph $G$ with the edge set $$E(G)={A\choose 1} \cup \left({[n]\choose 2}\setminus {A\choose 2}
\right).$$ It is easy to check $h_n(G)=\frac{5}{4}+O(\frac{1}{n})$ and that $G$ contains no copy of $K_{2}^{\{1,2\}}$.

Now we prove the upper bound. Consider any $K^{\{1,2\}}_2$-free hypergraph $G$ of edge-type $\{1,2\}$ on $n$ vertices. Let 