The dimension of a poset (P, ≤_P) is defined as the least cardinal λ such that there exists a λ-sized collection of linear extensions of P realizing P, that is to say a ≤_P b if and only a ≤ b in any linear extension in the collection. We will focus on the poset P_α(κ), that is the poset of subsets of κ of size less than α partially ordered by inclusion, and determine completely the dimension of such posets under GCH. Then we will mention a few consistency results when GCH fails. In particular, we point out the connection between the dimension of the poset P_α (2^κ) and the density of 2^κ under the <α-box product topology, and show it is consistent that they are different.