In many books I find these two definitions of Ramsey ultrafilters, extremely similar but different:
1)For every partition $\mathbb{N}=\bigsqcup A_k$ with $A_k\not\in\mathcal{U}$ there exists $X\in\mathcal{U}$ such that $|X\cap A_k|=1$.
2)For every partition $\mathbb{N}=\bigsqcup A_k$ with $A_k\not\in\mathcal{U}$ there exists $X\in\mathcal{U}$ such that $|X\cap A_k|\leq1$.
And in many proves these books pass from one of these definitions to the other without saying why they can do that. So I think they are equivalent, but I can't prove that despite I think it should be easy, otherwise the books should have spent some words about that.