A generalized Tur\'{a}n extension of the Deza--Erd{o}s--Frankl Theorem

For an integer r ≥ 3 and a subset L ⊂ [0,r-1], a graph G is (Kᵣ, 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 (Kᵣ, 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 ≤ |L| ≤ r-1, we establish the upper bound (1-1/3r) ∏∈ Ln-/r- for large n, which improves the bound provided by the celebrated Deza--Erd{o}s--Frankl Theorem by a factor of 1-1/3r. In the special case where L = , 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 Erd{o}s--Ko--Rado Theorem on t-intersecting systems to the generalized Tur\'{a}n setting. Our proof for the Deza--Erd{o}s--Frankl part involves an interesting combination of the Δ-system method and Tur\'{a}n's theorem. Meanwhile, for the Erd{o}s--Ko--Rado part, we employ the stability method, which relies on a theorem of Frankl regarding t-intersecting systems.