On the number of maximal independent sets and maximal induced bipartite subgraphs in $K_4$-free graphs

Let $G$ be a $K_4$-free graph of order $n$ and let $k$ be an integer with $0\leq k\leq n$. We show the existence of positive constants $η$ and $ν$ such that $G$ has at most $(4-η)^{(5-η)k-n}(5-η)^{n-(4-η)k}$ maximal independent sets of order $k$ and at most $O\left((12-ν)^{\frac{n}{4}}\right)$ maximal induced bipartite subgraphs.