  Low-energy degrees of freedom of a spin-1/2 kagome antiferromagnet in the
vicinity of the saturation field are mapped to a hard-hexagon model on a
triangular lattice. The latter model is exactly solvable. The presented mapping
allows to obtain quantitative description of the magnetothermodynamics of a
quantum kagome antiferromagnet up to exponentially small corrections as well as
predict the critical behavior for the transition into a magnon crystal state.
Analogous mapping is presented for the sawtooth chain, which is mapped onto a
model of classical hard dimers on a chain.
