  In the $\left(1:b\right)$ component game played on a graph $G$, two players,
Maker and Breaker, alternately claim~$1$ and~$b$ previously unclaimed edges of
$G$, respectively. Maker's aim is to maximise the size of a largest connected
component in her graph, while Breaker is trying to minimise it. We show that
the outcome of the game on the binomial random graph is strongly correlated
with the appearance of a nonempty $(b+2)$-core in the graph.
  For any integer $k$, the $k$-core of a graph is its largest subgraph of
minimum degree at least $k$. Pittel, Spencer and Wormald showed in 1996 that
for any $k\ge3$ there exists an explicitly defined constant $c_{k}$ such that
$p=c_{k}/n$ is the threshold function for the appearance of the $k$-core in
$G(n,p)$. More precisely, $G(n,c/n)$ has WHP a linear-size $k$-core when the
constant $c>c_{k}$, and an empty $k$-core when $c<c_{k}$.
  We show that for any positive constant $b$, when playing the $(1:b)$
component game on $G(n,c/n)$, Maker can WHP build a linear-size component if
$c>c_{b+2}$, while Breaker can WHP prevent Maker from building larger than
polylogarithmic-size components if $c<c_{b+2}$.
  For Breaker's strategy, we prove a theorem which may be of independent
interest. The standard algorithm for computing the $k$-core of any graph is to
repeatedly delete ("peel") all vertices of degree less than $k$, as long as
such vertices remain. When $G(n,c/n)$ for $c<c_{k}$, it was shown by Jiang,
Mitzenmacher and Thaler that $\log_{k-1}\log n+\Theta(1)$ peeling iterations
are WHP necessary and sufficient to obtain the (empty) $k$-core of~$G$. Our
theorem states that already after a constant number of iterations, $G$ is WHP
shattered into pieces of polylogarithmic size.
