Codegree-Constrained Turán Problems for $F_5$ and Its Blow-ups

Let $F_5$ denote the $3$-uniform hypergraph with edge set $\{abc,abd,cde\}$. For a hypergraph property $\mathcal{P}$, let $e{x}_{\mathcal{P}}(n,F_5)$ be the maximum number of edges in an $n$-vertex $F_5$-free $3$-uniform hypergraph satisfying $\mathcal{P}$. Let ${\mathrm{\Delta }}_2(H)$ denote the maximum codegree of a pair of vertices in a $3$-uniform hypergraph $H$. We determine the exact value of $e{x}_{{\mathrm{\Delta }}_2\ge m}(n,F_5)$ when $m$ is bounded away from $n\mathrm{/}3$ by a fixed positive proportion. More precisely, for every constant $c\in (1\mathrm{/}3,1)$, there exists $n_0$ such that for all $n\ge n_0$ and $m=cn$, $$e{x}_{{\mathrm{\Delta }}_2\ge m}(n,F_5)=m\lceil \frac{n-m}{2}\rceil \lfloor \frac{n-m}{2}\rfloor \mathrm{.}$$ The unique extremal hypergraph is the complete tripartite $3$-graph $K_{m,\lceil (n-m)\mathrm{/}2\rceil ,\lfloor (n-m)\mathrm{/}2\rfloor }^3$. We also prove the corresponding exact result for the blow-ups $F_5^t$, together with stability theorems showing that every near-extremal configuration is close to the relevant tripartite construction.