A Generalized Turán Extension of the Deza–Erdős–Frankl Theorem

Abstract

For an integer $r\ge 3$ and a subset $L\subset \left[0,r-1\right]$, a graph $G$ is $\left(K_r,L\right)$-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 $\left(K_r,L\right)$-intersecting graphs. The celebrated Ruzsa–Szemerédi Theorem corresponds to the case $r=3$ and $L=\left\{0,1\right\}$. For general $L$ with $2\le \mid L\mid \le r-1$, 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–Erdős–Frankl Theorem by a factor of $1-\frac{1}{3r}$. In the special case where $L=\left\{t,t+1,\dots ,r-1\right\}$, 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 $\Delta$-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.