  We report structural studies of the spin-orbit Mott insulator family
K$_x$Ir$_y$O$_2$, with triangular layers of edge-sharing IrO$_6$ octahedra
bonded by potassium ions. The potassium content acts as a chemical tuning
parameter to control the amount of charge in the Ir-O layers. Unlike the
isostructural families with Ir replaced by Co or Rh ($y=1$), which are metallic
over a range of potassium compositions $x$, we instead find insulating
behaviour with charge neutrality achieved via iridium vacancies, which order in
a honeycomb supercell above a critical composition $x_c$. By performing density
functional theory calculations we attribute the observed behaviour to a subtle
interplay of crystal-field environment, local electronic correlations and
strong spin-orbit interaction at the Ir$^{4+}$ sites, making this structural
family a candidate to display Kitaev magnetism in the experimentally unexplored
regime that interpolates between triangular and honeycomb structures.
