TURANS EXTREMAL PROBLEM IN RANDOM GRAPHS - FORBIDDING EVEN CYCLES

For 0 < gamma less than or equal to 1 and graphs G and H, we write G -->(gamma) H if any gamma-proportion of the edges of G span at least one copy of H in G. As customary, we write C-k for a cycle of length k. We show that, for every fixed integer l greater than or equal to 2 and real gamma > 0, there exists constant C-C(l, gamma) > 0 such that almost every random graph G(n,p) with p=p(n) greater than or equal to Cn(-1+1/(21-1)) satisfies G(n,p) -->(gamma) C-21. In particular, for any fixed 1 greater than or equal to 2 and gamma > 0, this result implies the existence of very sparse graphs G with G -->(gamma) C-21. (C) 1995 Academic Press, Inc.