\{F\ :\ i\notin F\in{\mathcal F}\}\ \ \ \text{and}\\
{\mathcal F}(i):=&\ \{F\setminus\{i\}\ :\ i\in F\in{\mathcal F}\}.\end{aligned}$$ Note that ${\mathcal F}(\bar i)\subset {X\setminus \{i\}\choose k}$ and ${\mathcal F}(i)\subset {X\setminus \{i\}\choose k-1}$.

Denote by $\mathcal E_{l}$ the maximal intersecting family with $|\mathcal E_l(\bar 1)|=l$, $|\bigcap_{E\in \mathcal E_l(\bar 1)}E|=k-1$ (note that the family is defined up to isomorphism). Note that $\mathcal J_2$ is isomorphic to $\mathcal E_2$.[^2] It is not difficult to see that for $k-1<l<n-k$ we have $\mathcal E_l\subset \mathcal E_{n-k}$: we have $\mathcal E_l(1) = \mathcal E_{n-k}(1)$ for this range. The following theorem is one of the main results in [@KostM]:

\[thmko\] Let $k\ge 5$ and $n=n(k)$ be sufficiently large. If ${\mathcal F}\subset {[n]\choose k}$ is intersecting and $|{\mathcal F}|> |\mathcal J_3|$ then ${\mathcal F}\subset \mathcal E_l$ for $l\in \{0,\ldots k-1, n-k\}$.

We note that it is easy to verify that $|\mathcal J_3|<|\mathcal E_{n-k}|$, e.g., for $n>4k$. The authors of [@KostM] we using the delta-systems method of Frankl.[^3] Actually, Theorem \[thmko\] can be deduced from the results of Frankl [@F16] with little extra effort.

Many results in extremal set theory are much easier to obtain once one assumes that $n$ is sufficiently large in comparison to $k$. (The possibility to apply the 