$c(n,k,4)$ and the structure of the extremal family for $n>n_0(k)$. The analysis in [@FOT1] is much more complicated than that in [@F16], and the problem for $\tau\ge 5$ is still wide open. The result of [@FOT1] may be extended to much smaller $n$ in a similar way, but, of course, progress on the case $\tau \ge 5$ would be more interesting.\
Erdős, Rothschild and Szemerédi (cf. [@Erd73]) raised the following question: how large can the intersecting family ${\mathcal F}\subset {[n]\choose k}$ be, given that $\Delta({\mathcal F})\le c|{\mathcal F}|$. They proved that, for any fixed $2/3<c<1$, there exists $n_0(k,c)$, such that for any $n>n_0(k,c)$ the largest intersecting family under this restriction is $\mathcal H_2$ (see ), up to isomorphism. Stronger results of this type were proven by Frankl [@F16] and by Füredi [@Fur]. See also a survey [@DeF], where the concise statement of the results is given.

The methods we developed in this paper allow us to extend these results to the range $n>Ck$, where $C$ depends on $c$ only. We are going to illustrate it for one such theorem, but we note that the same ideas would work for other cases. Let us state one of the theorems, proven in 