Biased positional games and small hypergraphs with large covers (Q1010790)

Summary: 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 \(b\leq(\log 2-o(1))n/\log n\). Also, in the biased \((1:b)\) avoider-enforcer game played on \(E(K_n)\), Enforcer can force Avoider to create a Hamilton cycle when \(b\leq (1-o(1))n/\log n\). These results are proved using a new approach, relying on the existence of hypergraphs with few edges and large covering number.