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{\bf J{\'o}zsef Balogh and Wojciech Samotij}
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{\bf On the Chv{\'a}tal-Erd{\H{o}}s Triangle Game}
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Given a graph $G$ and positive integers $n$ and $q$, let ${\bf G}(G;n,q)$
be the game played on the edges of the complete graph $K_n$ in which
the two players, Maker and Breaker, alternately claim $1$ and $q$
edges, respectively. Maker's goal is to occupy all edges in some copy
of $G$; Breaker tries to prevent it. In their seminal paper on
positional games, Chv{\'a}tal and Erd{\H{o}}s proved that in the game
${\bf G}(K_3;n,q)$, Maker has a winning strategy if $q < \sqrt{2n+2}-5/2$,
and if $q \geq 2\sqrt{n}$, then Breaker has a winning strategy. In
this note, we improve the latter of these bounds by describing a
randomized strategy that allows Breaker to win the game ${\bf G}(K_3;n,q)$
whenever $q \geq (2-1/24)\sqrt{n}$. Moreover, we provide additional
evidence supporting the belief that this bound can be further improved
to $(\sqrt{2}+o(1))\sqrt{n}$.
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