ABSTRACT For an integer and a subset , a graph is ‐intersecting if the number of vertices in the intersection of every pair of in belongs to . We study the maximum number of in an ‐vertex ‐intersecting graphs. The celebrated Ruzsa–Szemerédi Theorem corresponds to the case and . For general with , we establish the upper bound for large , which improves the bound provided by the celebrated Deza–Erdős–Frankl Theorem by a factor of . In the special case where , we derive the tight upper bound for large and establish a corresponding stability result. This is an extension of the seminal Erdős–Ko–Rado Theorem on ‐intersecting systems to the generalized Turán setting. Our proof for the Deza–Erdős–Frankl part involves an interesting combination of the ‐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 ‐intersecting systems.
Helliar et al. (Fri,) studied this question.