This paper aims at evaluating a novel multi-class methodology to establish Sleep Apnea-Hypopnea Syndrome (SAHS) severity by the use of single-channel at-home oximetry recordings. The study involved 320 participants derived to a specialized sleep unit due to SAHS suspicion. These were assigned to one out of the four SAHS severity degrees according to the apnea-hypopnea index (AHI): no-SAHS (AHI<5 events/hour), mild-SAHS (5 <= AHI<15 e/h), moderate-SAHS (15 <= AHI<30 e/h), and severe-SAHS (AHI >= 30 e/h). A set of statistical, spectral, and non-linear features were extracted from blood oxygen saturation (SpO(2)) signals to characterize SAHS. Then, an optimum set among these features were automatically selected based on relevancy and redundancy analyses. Finally, a multi-class AdaBoost model, built with the optimum set of features, was obtained from a training set (60%) and evaluated in an independent test set (40%). Our AdaBoost model reached 0.386 Cohen's kappa in the four-class classification task. Additionally, it reached accuracies of 89.8%, 85.8%, and 74.8% when evaluating the AHI thresholds 5 e/h, 15 e/h, and 30 e/h, respectively, outperforming the classic oxygen desaturation index. Our results suggest that SpO(2) obtained at home, along with multiclass AdaBoost, are useful to detect SAHS severity.