On set systems without singleton intersections

Abstract: Consider a family $\mathcal{F}$ of $k$-subsets of an ambient $(k<sup>2-k+1)$-set such that no pair of $k$-subsets in $\mathcal{F}$ intersects in exactly one element. In this short note we show that the maximal size of such $\mathcal{F}$ is $\binom{k<sup>2-k-1}{k-2}$ for every $k &gt; 1$.