can generalize this construction, giving the following lower bound for $S^k(K_2^{\{1,2\}})$ (the $k$-th suspension of $K_2^{\{1,2\}}$). The details of the computation are omitted. $$\label{eq:sk122}
    \pi(S^k(K_2^{\{1,2\}}))\geq 1+ \frac{1}{2^{k+1}}.$$

\[conj:1\] For any $k\geq 2$, $\pi(S^k(K_2^{\{1,2\}}))= 1+ \frac{1}{2^{k+1}}.$

Supersaturation and Blowing-up
==============================

Supersaturation Lemma [@ErSi] is an important tool for uniform hypergraphs. There is a natural generalization of the supersaturation lemma and blowing-up in non-uniform hypergraphs.

[**(Supersaturation)**]{} For any hypergraph $H$ and $a>0$ there are $b$, $n_0>0$ so that if $G$ is a hypergraph on $n>n_0$ vertices with $R(G)=R(H)$ and $h_n(G)>\pi(H)+a$ then $G$ contains at least $b{n\choose v(H)}$ copies of $H$.

[**Proof:**]{} Let $R:=R(H)$ and $r:=|R|$. Since $\displaystyle \pi(H)=\lim_{n\to\infty}\pi_n(H)$, there is an $n_0$ so that for $m\geq n_0$, $\pi_m(H)\leq \pi(H)+\frac{a}{2}$. For any $n_0\leq m\leq n$, there must be at least $\frac{a}{2r}{n\choose m}$ $m$-sets $M\subset V(G)$ inducing a $R$-graph $G[M]$ with $h(G[M])>\pi(H)+\frac{a}{2}$. Otherwise, we have $$\sum_{M}h_m(G[M])\leq \left(\pi(H)+\frac{a}{2}\right){n\choose m} + \frac{a}{2r}{n\choose m}r=(\pi(H)+a){n\choose m}.$$ But we also have $$\begin{aligned}
\sum_Mh_m(G[M]) &=\sum_M \sum_{\stackrel{F\in E(G)}{F\subseteq M}}\frac{1}{{m\choose |F|}} \\
                &=\sum_{F\in E(G)}\sum_{M\supseteq F}\frac{1}{{m\choose |F|}}  \\
                &= \sum_{F\in E(G)} \frac{{n-|F| \choose m-|F|}}{{m\choose |F|}}  \\
                &=\sum_{F\in E(G)} \frac{{n\choose m}}{{n\choose |F|}}\\
                &= {n\choose m}h_n(G).\end{aligned}$$ This is a contradiction to the assumption that $h_n(G)>\pi(H)+a$. By the choice of $m$, each of these $m$-sets contains 