Logic: upper limit for a minimum resolution of an unsatisfactory formula.

Leave $ varphi $ be a logical formula of the first order with $ n $ literals$ X_1, …, X_n $).

$ varphi $ It 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 $ The literals and we can solve them one by one. Some research taught me that there are unsatisfiable formulas that require exponentially many results for the empty clause. I read $ 4 ^ n $ somewhere, but I do not understand why this should be the upper limit.

I hope someone can help me solve this!