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Lower limit on the neutralino mass in the general MSSM

We discuss constraints on SUSY models with non-unified gaugino masses and R_P conservation. We derive a lower bound on the neutralino mass combining the direct limits from LEP, the indirect limits from gmuon, bsgamma, Bsmumu and the relic density constraint from WMAP. The lightest neutralino (mneutralino=6GeV) is found in models with a light pseudoscalar with MA<200GeV and a large value for $tanβ$. Models with heavy pseudoscalars lead to mneutralino>18(29)GeV for $\tanβ=50(10)$. We show that even a very conservative bound from the muon anomalous magnetic moment can increase the lower bound on the neutralino mass in models with mu<0 and/or large values of $\tanβ$. We then examine the potential of the Tevatron and the direct detection experiments to probe the SUSY models with the lightest neutralinos allowed in the context of light pseudoscalars with high $\tanβ$. We also examine the potential of an e+e- collider of 500GeV to produce SUSY particles in all models with neutralinos lighter than the W. In contrast to the mSUGRA models, observation of at least one sparticle is not always guaranteed.