Logic – Upper limit for a minimum resolution of the unsatisfactory formula

Let $ varphi $ to be a logical formula of first order with $ n $ literals ($ X_1, …, X_n $).

$ varphi $ is unsatisfiable.

Now, I want to know the upper limit of a minimum resolution of $ varphi $ resulting in the empty clause ($ square $).

My first idea was that it should be possible in linear time ($ n $) because the longest clause can have up to $ 2n $ literals and we can solve them one by one. Some research has learned that there are unsatisfactory formulas that require a considerable multiplication of results for the empty clause. I have read $ 4 ^ n $ somewhere but I do not understand why this should be the upper limit.

I hope someone can help me understand that!