Biased Positional Games and Small Hypergraphs with Large Covers - Krivelevich, Michael; Szabó, Tibor (The Electronic Journal of Combinatorics, 2008)
We prove that in the biased (1:b) Hamiltonicity and k-connectivity Maker-Breaker games (k>0 is a constant), played on the edges of the complete graph K_n, Maker has a winning strategy for b(2-o(1))n/n. Also, in the biased (1:b) Avoider-Enforcer game played on E(K_n), Enforcer can force Avoider to create a Hamilton cycle when b(1-o(1))n/n. These results are proved using a new approach, relying on the existence of hypergraphs with few edges and large covering number.