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We study the problem of detection of a p-dimensional sparse vector of parameters in the linear regression model with Gaussian noise. We establish the detection boundary, i.e., the necessary and sufficient conditions for the possibility of successful detection as both the sample size n and the dimension p tend to infinity. Testing procedures that achieve this boundary are also exhibited. Our results encompass the high-dimensional setting (p ≫ n). The main message is that, under some conditions,doi:10.1214/10-ejs589 fatcat:eadc5ulbincnpascq3232iaibq