Set Systems with No Singleton Intersection

Let $\mathcal{F}$ be a $k$-uniform set system defined on a ground set of size $n$ with no singleton intersection; i.e., no pair $A,B\in\mathcal{F}$ has $|A\cap B|=1$. Frankl showed that $|\mathcal{F}|\leq\binom{n-2}{k-2}$ for $k\geq4$ and $n$ sufficiently large, confirming a conjecture of Erdős and Sós. We determine the maximum size of $\mathcal{F}$ for $k=4$ and all $n$, and also establish a stability result for general $k$, showing that any $\mathcal{F}$ with size asymptotic to that of the best construction must be structurally similar to it.