The Gunners suffered a torrid first half as Robert Lewandowski’s header, Thomas Muller’s snapshot and David Alaba’s strike put the hosts 3-0 up. Arjen Robben’s first-time finish and a Muller breakaway strike came either side of Olivier Giroud’s volley. A record equalling European defeat leaves Arsenal six points behind Bayern and Olympiakos with two games left. The Greek side’s 2-1 win over Dinamo Zagreb, achieved via a 90th-minute winner, means that qualification is out of Arsenal’s hands and they are in danger of failing to reach the knockout stages for the first time in 13 seasons.

Mesut Ozil had a goal ruled out for handball when the score was 1-0 and Arsene Wenger’s team improved after the break. But Bayern’s relentless pressing and ability to pounce on defensive errors exposed the visitors, who lost for a third time in four Champions League group games. Arsenal centre-back Laurent Koscielny was ruled out with a hip injury hours before kick-off and his replacement, Gabriel, was partly at fault for two of Bayern’s three first-half goals. With a groin injury to right-back Hector Bellerin, the Gunners now have 10 players on the injury list. After defeats by Dinamo Zagreb and Olympiakos, the Gunners are attempting to become the first team since Galatasaray in 2012 to reach the knockout stages after losing their first two group games. Beating Bayern at home a fortnight ago gave the Gunners optimism but they must now make up a six-point gap to second-placed Olympiakos, who they face in their final group game on 9 December after hosting Zagreb on 24 November.

Arsenal need to beat Zagreb and hope Olympiakos to lose their next game, at Bayern, who are yet to qualify for the knockout stages. That would set up an intriguing final match in Greece where the Gunners would have to beat Olympiakos by two clear goals after conceding a sloppy winner in their 3-2 defeat at the Emirates on 29 September. Injuries and not signing the players needed have cost us.