Many triangles with few edges Extremal problems concerning the number of independent sets or complete subgraphs in a graph have been well studied in recent years. Cutler and Radcliffe proved that among graphs with n vertices and maximum degree at most r, where n  a(r  1)  b and 0 xe2x89xa4 b xe2x89xa4 r, aKr1 xe2x88xaa Kb has the maximum number of complete subgraphs, answering a question of Galvin. Gan, Loh and Sudakov conjectured that aKr1 xe2x88xaaKb also maximizes the number of complete subgraphs Kt for each fixed size t xe2x89xa53, and proved this for a  1. Cutler and Radcliffe proved this conjecture for r xe2x89xa4 6. We investigate a variant of this problem where we fix the number of edges instead of the number of vertices. We prove that aKr1 xe2x88xaaC(b), where C(b) is the colex graph on b edges, maximizes the number of triangles among graphs with m edges and any fixed maximum degree r xe2x89xa4 8, where m  a(r1 2 )  b and 0 xe2x89xa4 b u003c (r1 2 ). Mathematics Subject Classifications: 05.

Extremal Problems for tPartite and tColorable Hypergraphs Fix integers t ge r ge 2 and an runiform hypergraph F. We prove that the maximum number of edges in a tpartite runiform hypergraph on n vertices that contains no copy of F is c_t, Fn choose r o(nr), where c_t, F can be determined by a finite computation.   We explicitly define a sequence F_1, F_2, ldots of runiform hypergraphs, and prove that the maximum number of edges in a tchromatic runiform hypergraph on n vertices containing no copy of F_i is alpha_t,r,in choose r o(nr), where alpha_t,r,i can be determined by a finite computation for each ige 1. In several cases, alpha_t,r,i is irrational. The main tool used in the proofs is the Lagrangian of a hypergraph.