Set systems containing no singleton intersection and the Delsarte number

We prove that the maximum size of a family of $k$-element subsets of the set $[n] = \{1, 2, \ldots, n\}$ which contains no singleton intersection is $\binom{n-2}{k-2}$ when $3k-3 \le n \le k^2-k+1$. This improves upon a recent result of Cherkashin. Our proof uses Schrijver's variant of the Lov\'asz number and furnishes an infinite family of graphs where the Schrijver variant of the Lov\'asz number is strictly smaller than the Lov\'asz number. As a consequence of our result and a recent result of Keller and Lifshitz, it follows that for $k$ sufficiently large, the maximum size of a $k$-uniform family on $[n]$ containing no singleton intersection is $\binom{n-2}{k-2}$ for all $n\ge 3k-3$, which is the best possible threshold.