  Background: A newly identified dripline nucleus $^{31}$F offers a unique
opportunity to study the two-neutron ($2n$) correlation at the east shore of
the island of inversion where the $N = 28$ shell closure is lost.
  Purpose: We aim to present the first three-body theoretical results for the
radius and total reaction cross sections of $^{31}$F. This will further help to
investigate how the pairing and breakdown of the $N = 28$ shell closure
influence the formation of the $2n$-halo structure and the anti-halo effect in
this mass region.
  Methods: A $^{29}$F$+n+n$ three-body system is described by the cluster
orbital shell model, and its total reaction cross section is calculated by the
Glauber theory.
  Results: Our three-body calculations predict 3.48-3.70 fm for the
root-mean-square radius of $^{31}$F, which corresponds to the total reaction
cross section of 1530 (1410)-1640 (1500) mb for a carbon target at 240 (900)
MeV/nucleon. The binding mechanism and halo formation in $^{31}$F are
discussed.
  Conclusions: The present study suggests a novel anti-halo effect in this mass
region: When the pairing overcome the energy gap between the $p_{3/2}$ and
$f_{7/2}$ orbits, the inversion of the occupation number of these orbits takes
place, and it diminishes the $2n$-halo structure.
