Vertex-colored Turán theorems with applications in extremal hypergraph problems

Abstract

Balogh, Clemen, and Lidický proved that the ${\ell }_2$-norm Turán problem for $K_5^3$ is asymptotically solved by the balanced bipartite construction, and they further conjectured that this construction is uniquely extremal for all sufficiently large $n$. We confirm this conjecture. We also determine exactly the maximum number of cliques in an $n$-vertex $K_5^3$-free $3$-uniform hypergraph for all sufficiently large $n$, thereby verifying the corresponding case of a conjecture of Frankl, Gryaznov, and Talebanfard. The main ingredients are Turán-type theorems for vertex-colored graphs forbidding balanced cliques, including an edge bound, an ${\ell }_2$-norm bound, and a sharp crossing-triangle theorem in the two-colored balanced $K_4$-free case. We also use a local modification procedure within the stability method. This reduces the exact hypergraph problems to proving that the relevant objective function increases under suitable local changes near the bipartite construction.

34 pages, comments are welcome