On complete subgraphs of r-chromatic graphs

Let G,J n) be an r-chromatic graph with n vertices in each colour class . Suppose G = G 3 (n), and t (G) . the minimal degree in G, is at least n + t (t _> 1) . We prove that C contains at least t 3 triangles but does not have to contain more titan 4t 3 of them . Furthermore, we give lower bounds for s such that G contains a complete 3-partite graph with s vertices in each class . Let';.(ii) = max t6(G) : G = G i (n), G does not contain a complete graph with r vertices ; . We obtain various results on j 1 ht) . In particular, we prove that if cz, = lim ~ ~ (n )i tr, then Jim,_ (cr -(r 2)) > 112 and we conjecture that equality holds . We prove several other results and state a number of unsolved problems .