  We show that for an infinitely many natural numbers $k$ there are $k$-uniform
hypergraphs which admit a `rescaling phenomenon' as described in [9]. More
precisely, let $\mathcal{A}(k,I, n)$ denote the class of $k$-graphs on $n$
vertices in which the sizes of all pairwise intersections of edges belong to a
set $I$. We show that if $k=rt^2$ for some $r\ge 1$ and $t\ge 2$, and~$I$ is
chosen in some special way, the densest graphs in $\mathcal{A}(rt^2,I, n)$ are
either dominated by stars of large degree, or basically, they are `$t$-thick'
$rt^2$-graphs in which vertices are partitioned into groups of $t$ vertices
each and every edge is a union of $tr$ such groups. It is easy to see that,
unlike in stars, the maximum degree of $t$-thick graphs is of a lower order
than the number of its edges. Thus, if we study the graphs from
$\mathcal{A}(rt^2,I, n)$ with a prescribed number of edges $m$ which minimize
the maximum degree, around the value of $m$ which is the number of edges of the
largest $t$-thick graph, a rapid, discontinuous phase transition can be
observed. Interestingly, these two types of $k$-graphs determine the structure
of all hypergraphs in $\mathcal{A}(rt^2,I, n)$. Namely, we show that each such
hypergraph can be decomposed into a $t$-thick graph $H_T$, a special collection
$H_S$ of stars, and a sparse `left-over' graph $H_R$.
