On Colorings Avoiding a Rainbow Cycle and a Fixed Monochromatic Subgraph Let H and G be two graphs on fixed number of vertices. An edge coloring of a complete graph is called (H,G)good if there is no monochromatic copy of G and no rainbow (totally multicolored) copy of H in this coloring. As shown by Jamison and West, an (H,G)good coloring of an arbitrarily large complete graph exists unless either G is a star or H is a forest. The largest number of colors in an (H,G)good coloring of K_n is denoted maxR(n, G,H). For graphs H which can not be vertexpartitioned into at most two induced forests, maxR(n, G,H) has been determined asymptotically. Determining maxR(n; G, H) is challenging for other graphs H, in particular for bipartite graphs or even for cycles. This manuscript treats the case when H is a cycle. The value of maxR(n, G, C_k) is determined for all graphs G whose edges do not induce a star.

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.