A generalized Turán extension of the Deza--Erdős--Frankl Theorem

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–Szemerédi Theorem corresponds to the case $r=3$ and $L=\{0,1\}$. For general $L$ with $2\le \mathrm{\mid }L\mathrm{\mid }\le r-1$, we establish the upper bound $(1-\frac{1}{3r}){\prod }_{\mathrm{\ell }\in L}\frac{n-\mathrm{\ell }}{r-\mathrm{\ell }}$ for large $n$, which improves the bound provided by the celebrated Deza–Erdős–Frankl Theorem by a factor of $1-\frac{1}{3r}$. In the special case where $L=\{t,t+1,\dots ,r-1\}$, we derive the tight upper bound for large $n$ and establish a corresponding stability result. This is an extension of the seminal Erdős–Ko–Rado Theorem on $t$-intersecting systems to the generalized Turán setting. Our proof for the Deza–Erdős–Frankl part involves an interesting combination of the $\mathrm{\Delta }$-system method and Turán’s theorem. Meanwhile, for the Erdős–Ko–Rado part, we employ the stability method, which relies on a theorem of Frankl regarding $t$-intersecting systems.