Extremal Problems for t-Partite and t-Colorable Hypergraphs - Mubayi, Dhruv; Talbot, John (The Electronic Journal of Combinatorics, 2008)
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 c_t, Fn r +o(n^r), where c_t, F can be determined by a finite computation. We explicitly define a sequence F_1, F_2, … 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 F_i is α_t,r,in r +o(n^r), where α_t,r,i can be determined by a finite computation for each i1. In several cases, α_t,r,i is irrational. The main tool used in the proofs is the Lagrangian of a hypergraph.