  For an integer $r \ge 3$ and a subset $L \subset [0,r-1]$, a graph $G$ is
$(K_{r}, L)$-intersecting if the number of vertices in the intersection of
every pair of $K_r$ in $G$ belongs to $L$. We study the maximum number of $K_r$
in an $n$-vertex $(K_{r}, L)$-intersecting graphs. The celebrated
Ruzsa--Szemer\'{e}di Theorem corresponds to the case $r=3$ and $L = \{0,1\}$.
  For general $L$ with $2 \le |L| \le r-1$, we establish the upper bound
$\left(1-\frac{1}{3r}\right) \prod_{\ell \in L}\frac{n-\ell}{r- \ell}$ for
large $n$, which improves the bound provided by the celebrated
Deza--Erd\H{o}s--Frankl Theorem by a factor of $1-\frac{1}{3r}$. In the special
case where $L = \{t, t+1, \ldots, r-1\}$, we derive the tight upper bound for
large $n$ and establish a corresponding stability result. This is an extension
of the seminal Erd\H{o}s--Ko--Rado Theorem on $t$-intersecting systems to the
generalized Tur\'{a}n setting.
  Our proof for the Deza--Erd\H{o}s--Frankl part involves an interesting
combination of the $\Delta$-system method and Tur\'{a}n's theorem. Meanwhile,
for the Erd\H{o}s--Ko--Rado part, we employ the stability method, which relies
on a theorem of Frankl regarding $t$-intersecting systems.
