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{\bf Michael Krivelevich and Tibor Szab\'o}
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{\bf Biased Positional Games and Small Hypergraphs with Large Covers}
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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.



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