On the Tur\'an number of blow-ups of $\mathcal{F}_5$
Let $\mathcal{F}_5$ denote the $3$-uniform hypergraph on the vertex set $\{f_1,f_2,\dots,f_5\}$ with hyperedges $\{f_1f_2f_3,f_1f_2f_4,f_3f_4f_5\}$. Recently, Balogh, Clemen and Luo determined the Tur\'an number of a one-vertex blow-up of $\mathcal{F}_5$, more specifically, they blow up the vertex $f_5$ to $t$ vertices, the resulting hypergraph is denoted by $\mathcal{F}_5(f_5;t)$. They show that for infinitely many $t$, $\mathcal{F}_5(f_5;t)$ has exponentially many extremal constructions and positive Tur\'an density. In this paper, we determine the exact Tur\'an number of the hypergraph obtained by blowing up $f_3$ of $\mathcal{F}_5$ to $t$ vertices and show that it also has exponentially many extremal constructions. We also give a general upper bound and lower bound of the Tur\'an number of every blow-up of $\mathcal{F}_5$. For some special blow-ups of $\mathcal{F}_5$, for example, $t$-disjoint copies of $\mathcal{F}_5$, we determine the exact Tur\'an number. We construct a hypergraph $\mathcal{F}_{sim}(t)$ which is a subgraph of a blow-up of $\mathcal{F}_5$, and is contained in the hypergraph obtained by adding any new hyperedge to the Tur\'an hypergraph (the balanced complete $3$-partite hypergraph), but its extremal construction is not the Tur\'an hypergraph. We also determine the exact Tur\'an number of $\mathcal{F}_{sim}(t)$.