Biased Positional Games and Small Hypergraphs with Large Covers

Abstract

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