Almost Intersecting Families

Let $n > k 1$ be integers, $[n] = \{1, \ldots, n\}$. $\mathcal F$ a family of $k$-subsets $[n]$. The is called intersecting if $F \cap F' \neq \varnothing$ for all $F, \in \mathcal F$. It almost it not but to every there at most one $F'\in satisfying \varnothing$. Gerbner et al. proved that \geq 2k + 2$ then $|\mathcal F| \leqslant {n - 1\choose 1}$ holds families. Our main result implies the considerably stronger and best possible bound 1} (2 o(1))k$, $k\ge 3$.