[@F16]. Recall that a (projective plane of order $2$) $\mathcal P$ is a family consisting of $7$ $3$-element sets $P_1,\ldots, P_7$, such that $|P_i\cap P_j|=1$ for each $i\ne j$.

\[thmfano\] Suppose that ${\mathcal F}\subset {[n]\choose k}$ is intersecting, and $\Delta({\mathcal F})\le c|{\mathcal F}|$ for some $c\in (3/7,1/2)$. Assume that ${\mathcal F}$ has the largest cardinality among such families. Then, for any $n>n_0(c,k)$, ${\mathcal F}$ is isomorphic to $$\mathcal D_{3/7}:=\ \Big\{F\in {[n]\choose k}\ :\  P\subset F\text{ for some }P\in \mathcal P\Big\}.$$

Our contribution here is as follows.

\[thmbounddeg\] There exists an absolute constant $C$, such that the following holds. In terms of Theorem \[thmfano\], if there exists $\epsilon>0$, such that $c\in [3/7+\epsilon, 1/2-\epsilon]$, then the conclusion of Theorem \[thmfano\] holds for $n>\frac{C}{\epsilon}k$.

The proof of this theorem is given in Section \[sec52\]. In Section \[sec6\], we discuss our results and methods, as well as pose some open problems.

The complete diversity version of Frankl’s theorem {#sec3}
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Let us begin with the statement of the Kruskal–Katona theorem. We shall state the Kruskal–Katona theorem in two different forms, but the most handy for our purposes is the form due to Hilton in terms of cross-intersecting families. Let us first give some definitions.

