is hidden behind the binomial coefficients in the formulation. Thus, later in the section (and as an intermediate step of the proof), we shall give a “conceptual” version of our main theorem. We note that the proof that we present is completely computation-free. In Sections \[sec33\], \[sec34\] we present strengthenings and generalisations of our main result. The cases of equality in Theorems \[thmfull1\] and \[thmfull2\] are described in Theorem \[thmfulleq\]. We provide a weighted version of the main result and a generalization to the case of cross-intersecting families.

We note that the main results of the section are meaningful for any $k\ge 3$. (This is by no means a serious restriction since possible structure of intersecting families in ${[n]\choose k}$ for $k\le 2$ is trivial.)+0.1cm

The following representation of natural numbers is important for the (classic form of) the Kruskal–Katona theorem. Given positive integers $\gamma$ and $k$, one can always write down $\gamma$ [*uniquely*]{} in the : $$\gamma = {a_k\choose k}+{a_{k-1}\choose k-1}+\ldots +{a_s\choose s}, \ \ a_k>a_{k-1}>\ldots >a_s\ge 1.$$

For the sake of comparison, let us state the classical version of the Kruskal–Katona theorem (equivalent to Theorem \[thmHil\]).

\[thmkk\] Let ${\mathcal F}\subset {[n]\choose k}$ and put $$\partial({\mathcal F}):=\big\{F'\in {[n]\choose k-1}\ 