  Three body resonant interactions between Rydberg atoms are considered in
order to perform few-body quantum gates. So far, the resonances found in cesium
or rubidium atoms relied on an adjacent two-body resonance which ceases to
exist for principal quantum numbers above $n \simeq 40$. We have proposed
recently a new class of 3-body interaction resonances in alkali-metal Rydberg
atoms [P. Cheinet \textit{et al.}, Quant. Elect. \textbf{50}, 213 (2020)],
which circumvienes this limit. We investigate here the relative strength
between this new class of 3-body interaction resonance and quasi-forbidden
2-body interaction resonances in rubidium and cesium Rydberg atoms. We then
identify the best case scenario for detecting and using this 3-body
interaction.
