  The random geometric graph is obtained by sampling $n$ points from the unit
square (uniformly at random and independently), and connecting two points
whenever their distance is at most $r$, for some given $r=r(n)$. We consider
the following variation on the random geometric graph: in each of $n$ rounds in
total, a player is offered two random points from the unit square, and has to
select exactly one of these two points for inclusion in the evolving geometric
graph. We study the problem of avoiding a linear-sized (or "giant") component
in this setting. Specifically, we show that for any $r\ll(n\log\log n)^{-1/3}$
there is a strategy that succeeds in keeping all component sizes sublinear,
with probability tending to one as $n\to\infty$. We also show that this is
tight in the following sense: for any $r\gg(n\log\log n)^{-1/3}$, the player
will be forced to create a component of size $(1-o(1))n$, no matter how he
plays, again with probability tending to one as $n\to\infty$. We also prove
that the corresponding offline problem exhibits a similar threshold behaviour
at $r(n)=\Theta(n^{-1/3})$. These findings should be compared to the existing
results for the (ordinary) random geometric graph: there a giant component
arises with high probability once $r$ is of order $n^{-1/2}$. Thus, our results
show, in particular, that in the geometric setting the power of choices can be
exploited to a much larger extent than in the classical Erd\H{o}s-R\'{e}nyi
random graph, where the appearance of a giant component can only be delayed by
a constant factor.
