Someone else gave a web link, and I assume in general you can Google for more as well as I can.
For books, "With the Yanks in Korea Vol 1" by Brian Cull and Dennis Newton (there is unfortunately no other vol., nor likely to be any time soon per Mr. Cull) deals with all air ops in Korea through end 1951, but with emphasis on the Commonwealth air units, including 77 sdn RAAF's Meteors, which flew the a/c from summer 1951 (after converting from Mustangs) to the end of the war. A first hand account from 77 sdn is "Escape from North Korea" by Ron Guthrie.
From the above plus ex-Soviet and Chinese sources, here's a summary of Meteor air-air encounters in Korea. All opposing a/c were MiG-15's. All opposing units were Soviet AF except as noted. 'Claims' means claimed destroyed except as noted:
August 25, 1951: In their first combat with MiG-15's the Meteors make no claim nor suffer any losses. The 17th Fighter Regiment (Russian acronym IAP) claims 2 Meteors with no loss.
Aug 29 1951: Meteors (Guthrie) claim one MiG 'possible', but Guthrie is shot down and serves out the war as POW. The 18th Guards Fighter Regiment (Russian acronym GIAP) claimed 2 Meteors with no loss.
Sep 5 '51: Meteors make no claim, one Meteor damaged by MiG (A77-726 returned to duty in Oct). The 523rd IAP claims 3 Meteors w/o loss.
Sep 10: Meteors no claims or losses. 176th GIAP claims one Meteor.
Sep 26: Meteors claim one MiG damaged, one Meteor damaged (A77-949, not seriously). 523rd and 17th IAP each claim one Meteor, w/o loss.
Sep 27: Meteors make no claim, one Meteor damaged (A77-744, not seriously). There is no known Soviet claim, possibly PLAAF, whose units re-entered Korean combat (after a brief stint winter of 50-51) at this time.
Oct 24, '51: Meteors make no claim, one Meteor damaged by MiG (A77-316, repaired, lost in 1952). The 523rd IAP claims again 3 Meteors, and the 18th GIAP another.
Nov 2 '51: Meteors claim MiG-15 damaged without loss. The 18th GIAP claimed 1 Meteor destroyed without loss.
Nov 3: Meteors claim MiG-15 damaged, one Meteor damaged (A77-373, repaired, lost '52). 523rd IAP claims 2 Meteors w/o loss.
Nov 7: Meteors no claims or losses. 176th GIAP claims one Meteor.
Dec 1 '51: Meteors claim 2 MiG's destroyed but lose 3 Meteors. The 176 GIAP claims 9 Meteors w/o loss.
Not long after, 77 sdn was shifted back to fighter bomber duty. It was apparent the Meteor could not deal on equal terms with the numerous aggressively piloted MiG's (though not known specifically to be regular Soviet AF units), particularly in high altitude fighter sweep/escort situations. The Meteors had lost 4 while claiming 2 MiG's. All but possibly one encounter was with the Soviets, who by their calculations had downed no fewer than 27 Meteors, without loss, up to December 1 1951.
In fighter bomber ops the Meteors had some more MiG encounters:
May 4, 1952: Meteors claim one MiG probable, the opponent and outcome is not known, but not Soviet MiG's.
May 8: Again Meteors claim one MiG probable w/o loss. A Chinese official source claims the PLAAF 45th Fighter Regiment destroyed 3 Meteors but lost 2 MiG's in a combat on this date: the Meteor's only known air-air success in Korea.
Oct 2 '52: Meteor (piloted by RAF F/O OM Cruikshank) is downed by MiG. The 518th IAP claims one Meteor.
March 27, 1953: Meteors claim one MiG probable with one Meteor damaged. The opponent and outcome not known but again not Soviet.
Altogether the PLAAF claimed 8 Meteors during the war, but I don't know the individual units/dates. Apparently they did not actually down any.
Altogether Meteors in Korea suffered 5 air combat losses, all against the Soviets, and are known to have downed 2 MiG's, both Chinese.
Joe