Both are ok. It really depends on what you want to say. A day means one day after the game.
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A day after the game, players took a long-expected vacation.In the context, if you say "a day after the game", then you are referring to an amount of time. In other words, you would be telling the reader how long after the game the players went on vacation. It would be similar to saying this:
IvanhrHowever, 'a day after the game' refers to a specific point in time just like the day after the game.In the context, "a day" refers only to an amout of time (i.e. how much time). The use of "the day" refers to a specific point in time (i.e. when).