A generalized Tur\'an extension of the Deza--Erdos--Frankl Theorem

For an integer r 3 and a subset L 0, r-1, a graph G is (Kₑ, L) -intersecting if the number of vertices in the intersection of every pair of Kᵣ in G belongs to L. We study the maximum number of Kᵣ in an n-vertex (Kₑ, L) -intersecting graphs. The celebrated Ruzsa--Szemer\'edi Theorem corresponds to the case r=3 and L = \0, 1\. For general L with 2 |L| r-1, we establish the upper bound (1-13r) ₋n-r- for large n, which improves the bound provided by the celebrated Deza--Erdos--Frankl Theorem by a factor of 1-13r. In the special case where L = \t, t+1, , r-1\, we derive the tight upper bound for large n and establish a corresponding stability result. This is an extension of the seminal Erdos--Ko--Rado Theorem on t-intersecting systems to the generalized Tur\'an setting. Our proof for the Deza--Erdos--Frankl part involves an interesting combination of the -system method and Tur\'an's theorem. Meanwhile, for the Erdos--Ko--Rado part, we employ the stability method, which relies on a theorem of Frankl regarding t-intersecting systems.