Arsenal can still qualify for European football next season if they manage to finish either 6th or 7th on the Premier League table.
Mathematically the Gunners can still qualify for the Europa League next season if luck is somehow in their favour and they manage to finish 6th. Finishing 7th on the table means they would qualify for the new third tier of European football, the Europa Conference League.
Arsenal currently sit 9th on the table with two games left (Crystal Palace & Brighton). Before we begin the permutations we assume that they win these games and reach 61 points, currently on 55.
Above Arsenal are Everton 56, West Ham 59 and Spurs 59 points. If both West Ham and Tottenham manage to win their next games, Arsenal will have absolutely no chance of qualifying for Europe.
So for Arsenal to have a chance they will to hope results go in their favour and the teams they chase slip up.
Everton who are 8th have to play Wolves and Man City, it’s unlikely that they’d win both games pushing Arsenal to 8th position.
Now for Arsenal to get to 7th position either Tottenham or West Ham will have to fail to win their remaining games.
Both are on 59 points with Tottenham having the superior goal difference. Tottenham play Aston Villa and Leicester while West Ham play West Brom and Southampton.
If results miraculously go Arsenal’s way and either West Ham or Tottenham fail to win their last games, Arsenal would sneak into Europe.