By the Erdos-Ko-Rado theorem, a $k$-uniform intersecting family $\mathcal{F}$ of $n$ elements has order at most $\binom{n-1}{k-1}$. Assuming $n\gneq 2k$, we have equality if and only if $\mathcal F$ is a star.
However the problem that I am facing is that given $n>2k$ and a $k$-uniform intersecting family $\mathcal F$ on $[n]$ (the first $n$ positive integers) s.t any element of $[n]$ is contained in at least $\binom{n-2}{k-2}$ sets of $\mathcal F$, then we have $\mathcal F$ is the boundary case above, i.e. a star. You can't simply double count to obtain the order of $\mathcal F$, since that would give an answer strictly lower than $\binom{n-1}{k-1}$.
I tried to prove the case of $k=2$ using graphs and succeeded, but I can't generalize it to higher $k$'s. I also tried induction but still failed. I think it has something to do with the 'unevenness' of the distribution of sets containing elements of $[n]$ near boundary case. Can someone help me out?
Edit: Still haven't solved this after hours of thinking.