Extremal Problems for t-Partite and t-Colorable Hypergraphs

Fix integers t ≥ r ≥ 2 and an r-uniform hypergraph F . We prove that the maximum number of edges in a t-partite r-uniform hypergraph on n vertices that contains no copy of F is ct,F ( n r ) + o(nr), where ct,F can be determined by a finite computation. We explicitly define a sequence F1, F2, . . . of r-uniform hypergraphs, and prove that the maximum number of edges in a t-chromatic r-uniform hypergraph on n vertices containing no copy of Fi is αt,r,i ( n r ) + o(nr), where αt,r,i can be determined by a finite computation for each i ≥ 1. In several cases, αt,r,i is irrational. The main tool used in the proofs is the Lagrangian of a hypergraph.