Small edge supports in a high-girth hypergraph

Let $e_0,\dots ,e_{s-1}$ and $x_0,\dots ,x_{s-1}$ form a Berge cycle with all $e_i\in \mathcal{F}$. The alternating sequence $(e_0,1),(x_0,0),(e_1,1),(x_1,0),\dots ,(e_{s-1},1),(x_{s-1},0),(e_0,1)$ is a closed walk in $I_{\mathcal{F}}$: each consecutive pair is incident, and every displayed node belongs to the selected vertex set. The hyperedge nodes are distinct, the vertex nodes are distinct, and the tags separate the two kinds. Thus it is a graph cycle of length $2s$. This includes $s=2$, where the four tagged nodes are distinct.

Conversely, every edge of $I_{\mathcal{F}}$ joins a vertex node to a hyperedge node. A cycle in it therefore alternates between the two kinds. Its length is even and at least four, say $2s$ with $s\ge 2$. Starting at a hyperedge node and choosing a direction gives the sequence in step 1.1; distinctness of the graph-cycle nodes makes the $e_i$ distinct and the $x_i$ distinct, while its edges give $x_i\in e_i\cap e_{i+1}$ cyclically. All $e_i$ lie in $\mathcal{F}$, so these data form a Berge cycle of length $s$. The two constructions recover the same cyclic sequence, apart from starting point and direction.

If $I_{\mathcal{F}}$ had a cycle, step 2.1 would give a Berge cycle of length $s$ with $s$ distinct hyperedges in $\mathcal{F}$. Hence $s\le \mathrm{\mid }\mathcal{F}\mathrm{\mid }<q$, contradicting the assumed Berge girth. Thus $I_{\mathcal{F}}$ has no cycle and is a forest; isolated selected vertices cannot create a cycle.