Frederik Pohl's Star Science Fiction was the first series of original anthologies, debuting in 1953 from Ballantine Books, with a first volume that included stories by C.M. Kornbluth, Fritz Leiber, Clifford D. Simak, Ray Bradbury, Issac Asimov, and Arthur C. Clarke ("The Nine Billion Names of God"). Three volumes, plus an interstitial volume of Short Novels, were published both in hardcover and paperback in 1953 and 1954; after a lapse, a single issue of Star Science Fiction Magazine and three further anthologies published in paperback only appeared in 1958 and 1959.
(The photo shows a first edition of Short Novels within a uniform set of November 1972 paperback reprints of the six Stars with wrap-around cover art by John Berkey.)
The first Star Science Fiction Stories was among the earliest titles published by Ballantine Books, created in 1953 by Ian and Betty Ballantine. Among Ian Ballantine's innovations in his new company was simultaneous hardcover/paperback publication of many of his titles (not all of them SF), and this series of SF anthologies distinguished from the competition by containing original, not reprint stories. Pohl, an active agent for many prominent SF authors at the time (and an author himself; The Space Merchants was another early Ballantine title), was in a perfect position to edit the series.
Early volumes published classic stories by Isaac Asimov, Alfred Bester, Jerome Bixby ("It's a Good Life," basis for the famous Twilight Zone episode), Ray Bradbury, Lester del Rey, Fritz Leiber, and others. The later volumes, a bit thinner, included notable stories by C.M. Kornbluth, James E. Gunn, Chan Davis, and Cordwainer Smith.
Star Science Fiction Stories, (Ballantine, 1953)
[isfdb]
Star Science Fiction Stories No. 2, (Ballantine, 1954)
[isfdb]