500 MeV generating very strong binding of the meson and a strong contraction of the few-nucleon systems. However, to reproduce the KEK data, these calculations involved some relaxation of the NN repulsion at short distances which would allow the existence of strongly bound and very dense systems. These calculations raise the important question on how to implement a realistic short range NN repulsion in the kaonic systems.

Another open question is related to the strength and range of KN interactions. Any mathematical description of few body systems requires knowledge of NN and KN off-shell scattering amplitudes. Those related to NN interactions are controlled fairly well in terms of modern NN potentials. For a bound K-meson the amplitudes needed involve the subthreshold energy region $$\label{1}
 f_{KN}=  f_{KN}( - E_B - E_{\rm{recoil}}),$$ where $E_B$ is the KN separation energy and $E_{\rm{recoil}} $ the recoil energy of the KN pair relative to the rest of the system. If the separation energy is as large as 100 MeV, meson momenta become $\approx 250 $ MeV/c and $ E_{\rm{recoil}}$ may be as large as $40$ MeV. The energies of interest for $ (- E_B -
E_{\rm{recoil}})$ are then located well below the $\Lambda(1405)$ state. The 