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Arsenal lost their appeal against Max Dowman's red card for violent conduct in their Carabao Cup exit to Fleetwood Town by two votes to one.
The 16-year-old was sent off in the 86th minute at Highbury for kicking out at Harrison Neal in retaliation for a strong tackle.
The Gunners submitted two videos as evidence to the Football Association's regulatory commission, along with a letter from secretary Zayna Perkins.
The letter set out in detail two grounds for the appeal - that it was wrong and that a three-game ban was "clearly excessive".
The majority view accepted that Dowman had received a forceful challenge that he did not see coming.
But the two commission members felt Dowman had the option to "step out of this challenge without any reaction but he turned back towards Neal and his forceful attempt at a retaliatory kick gave the referee a decision to make".
The minority view was that Dowman's actions were petulant rather than an act of violent conduct, and "did not constitute a challenge made with excessive force or brutality".
Arsenal's appeal that the ban was "clearly excessive" was unanimously rejected by the commission members, who were Udo Onwere, Stuart Ripley and Ray Olivier.
The commission noted that the threshold for a "clearly excessive" appeal is a high bar to reach.
Dowman's age was rejected as a relevant factor in deliberations and the commission could find no "truly exceptional" reason to alter the suspension.
The written reasons also revealed that the decision to dismiss Dowman was taken by referee Michael Salisbury, rather than on the advice of another member of the officiating team.
Dowman missed Arsenal's 1-0 win at Liverpool on Sunday, and will also sit out the next two Premier League games against Hull City and Newcastle United.
Fleetwood won the game 5-3 on penalties after a 1-1 draw.