allows us to prove Theorem \[thmclass2\] is that this bipartite switching is possible even in situations when we know practically nothing about the structure of the family.\
In the remaining part of the introduction, we present two results that extend some classic results of Frankl, Erdős Rothshild and Semeredi, and Furedi proven for $n>n_0(k)$ (with double-exponential dependency on $k$, coming from the aforementioned delta-method) to the range $n>Ck$, where $C$ is an absolute constant. Apart from the bipartite switching trick, we use as a main tool the junta approximation theorem due to Dinur and Friedgut [@DF]. The approach is resemblant of the recent paper [@Kup21] of the author, but here we apply it to a much wider class of problems. The rough framework of combining junta method to get approximate structure with combinatorial arguments for finer structure was used in an excellent recent paper [@KL]. One novel aspect in the use of junta method in the first problem below is that the actual extremal configuration is quite far from being a junta (i.e., the family is not defined by the intersection with a constant-size subset of the ground set, see precise definition in Section \[sec5\])! This, of course, poses additional 