  The extreme frustration in the Kagome antiferromagnet makes it possible to
realize a large number of closely competing magnetic phases either at the
classical or the quantum level. Motivated by recent neutron scattering study on
the Kagome antiferromagnet vesignieite BaCu$_{3}$V$_{2}$O$_{8}$(OH)$_{2}$, we
have made systematic investigation of the phase diagram of the classical and
quantum $J_{1}-J_{3}$ antiferromagnetic Heisenberg model defined on the Kagome
lattice(KAFHM). While it is shown previously that a large antiferromagnetic
exchange between the third-neighboring spins can drive an emergent $q=4$ Potts
order through the order-by-disorder mechanism, it is elusive what will happen
in the so called "grey region" where the Luttinger-Tisza criteria fails to
predict the classical ordering pattern. Through extensive Monte Carlo
simulation, we find that the "grey region" is characterized by competing
emergent $q=3$ and $q=4$ Potts order, whose emergence are beyond the
description of the conventional order-by-disorder mechanism. Our Schwinger
Boson mean field calculation in the "grey region" confirms the existence of the
Potts-3 order in the ground state of quantum $J_{1}-J_{3}$ KAFHM. The predicted
Potts-3 state is found to take two different forms distinguished by their
projective symmetry group character(PSG or quantum order), although they have
exactly the same symmetry. A comparison with the exact diagonalization result
on a 36-site cluster shows that the "grey region" may indeed host a nematic
spin liquid ground state featuring an anisotropic ring structure around the
$\mathbf{q}=0$ point in its spinon dispersion relation. We find that such "grey
region" of extremely frustrated magnets should better be taken as the
playground to study the rich competition between exotic emergent phases, rather
than a burden on their theoretical analysis.
