size $c(n,k,t)$ of the largest intersecting family ${\mathcal F}\subset {[n]\choose k}$ with $\tau({\mathcal F})\ge t$? Define $\mathcal C_3(n,k)\subset {[n]\choose k}$ to be the maximal intersecting family with $\mathcal C_3(n,k)(\bar 1)$ isomorphic to $\mathcal T_2(k)$. It is easy to see that $\tau(\mathcal C_3(n,k))=3$. Frankl managed to prove the following theorem.

\[thmtau3\] Let $k\ge 3$ and $n\ge n_0(k)$. Then $c(n,k,3)= |\mathcal C_3(n,k)|$. Moreover, for $k\ge 4$ the equality holds only for families isomorphic to $\mathcal C_3(n,k)$.

As in the case of Theorem \[thmko\], the bound on $n$ is doubly exponential in $k$. One of the main results of this paper is the proof of Theorem \[thmtau3\] under much milder restrictions.

\[thmtau3\] The conclusion of Theorem \[thmtau3\] holds for any $n>Ck$, where $C$ is an absolute constant, independent of $k$.

An important tool in the proof (as well as in the proof of Theorem \[thmfull2\]) is Lemma \[lemmin\], in which we found an elegant way to bound the sizes of families ${\mathcal F}$ satisfying $\tau({\mathcal F})=3$ and $\tau({\mathcal F}(\bar 1))=2$, and where ${\mathcal F}(\bar 1)$ is minimal w.r.t. this property.

In the paper [@FOT1], the authors managed to extend the result of [@F16] to the case $\tau =4$, determining the exact value of 