  We show that Vojta's conjecture for some rational surfaces is related to the
$abc$ conjecture. More specifically, we prove that Vojta's conjecture on these
surfaces implies a special case of the $abc$ conjecture, while the $abc$
conjecture implies Vojta's conjecture on these surfaces. Moreover, for similar
but different rational surfaces, we prove Vojta's conjecture unconditionally.
To prove these results, we use some (possibly new) properties of Farey
sequences.
