Biased positional games for which random strategies are nearly optimal

For a graph G and natural numbers n and q let G(G; n, q) be the game on the complete graph K-n in which two players, Maker and Breaker, alternately cl aim 1 and q edges respectively. Maker's aim is to build a copy of G while B reaker tries to prevent it. Let m(G)= max {e(H)-1/v(H)-2 : H subset of or e qual to G, v(H) greater than or equal to 3}. It Is shown that there exist c onstants c(0) and C-0 such that Maker has a winning strategy in G(G;n,q) if q less than or equal to c0n(1/m(G)), while for q greater than or equal to Con(1/m(G)) the game can be won by Breaker.