Turán problems for integer-weighted graphs

A multigraph is (k, r)-dense if every k-set spans at most r edges. What is the maximum number of edges exN(n, k, r) in a (k, r)-dense multigraph on n vertices? We determine the maximum possible weight of such graphs for almost all k and r (e.g., for all r > k) by determining a constant m = m(k, r) and showing that exN(n, k, r) = m ( n 2 ) +O(n), thus giving a generalization of Turán’s theorem. We find exact answers in many cases, even when negative integer weights are also allowed. In fact, our main result is to determine the maximum weight of (k, r)-dense n-vertex multigraphs with arbitrary integer weights with an O(n) error term. c © 2/11/1999 rev 2/27/01, 8/25/01 John Wiley & Sons, Inc. Journal of Graph Theory Vol. SUBMITTED, 1 31 (2/11/1999 rev 2/27/01, 8/25/01) c © 2/11/1999 rev 2/27/01, 8/25/01 John Wiley & Sons, Inc. CCC ??? 2 JOURNAL OF GRAPH THEORY