Collins Russian Dictionary 2nd Edition © HarperCollins Publishers 2000, 1997:
cotton [ˈkɔtn] n (fabric) хло́пок* хлопчатобума́жная ткань f (plant) хлопча́тник; (thread) (шве́йная) ни́тка#
cotton dress, etc хлопчатобума́жное пла́тье* etc
cotton on vi (inf)
he has cottoned on to the fact that ... до него́ дошло́, что ...
* is used to mark translations which have irregular inflections. The Russian-English side of the dictionary gives inflectional information.