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Abstract: We prove alower bound of ( 1 ln 1 + VCdim(C) ) on the number of random examples required for distribution-free learning of a concept class C, where VCdim(C) is the Vapnik-Chervonenkis dimension and and are the accuracy and con dence parameters. This improves the previous best lower bound of ( 1 ln 1 +VCdim(C)), and comes close to the known general upper bound of O ( 1 ln 1 + VCdim(C) ln 1) for consistent algorithms. We show that for manyinteresting concept classes, including kCNF and kDNF, our bound is actually tight to within a constant factor.