  Let $K$ be a field whose absolute Galois group is finitely generated. If $K$
neither finite nor of characteristic 2, then every hyperelliptic curve over $K$
with all of its Weierstrass points defined over $K$ has infinitely many
$K$-points. If, in addition, $K$ is not locally finite, then every elliptic
curve over $K$ with all of its 2-torsion rational has infinite rank over $K$.
These and similar results are deduced from the Hales-Jewett theorem.
