A note on $G$-intersecting families

Consider a graph $G$ and a $k$-uniform hypergraph $\mathcal{H}$ on common vertex set $[n]$. We say that $\mathcal{H}$ is $G$-intersecting if for every pair of edges in $X,Y\in \mathcal{H}$ there are vertices $x\in X$ and $y \in Y$suchthat$x = y$or$x$and$y$arejoinedbyanedgein$G$. This notion was introduced by Bohman, Frieze, Ruszink'o and Thoma who proved a natural generalization of the Erd\H{o}s-Ko-Rado Theorem for $G$-intersecting $k$-uniform hypergraphs for $G$ sparse and $k=O(n^{1\mathrm{/}4})$. In this note, we extend this result to $k=O\left(\sqrt{n}\right)$.