Given that \(S \subset T \subset W\), let's analyze each statement one by one using the given information:

I. \(S \cap T \cap W = S\):

This statement means that the intersection of sets \(S\), \(T\), and \(W\) is equal to set \(S\). Since \(S\) is a subset of both \(T\) and \(W\), the intersection of these sets would indeed be \(S\). This statement is true.

II. \(S \cup T \cup W = W\):

This statement means that the union of sets \(S\), \(T\), and \(W\) is equal to set \(W\). Since \(S\) is a subset of \(W\) and \(T\) is a subset of \(W\), the union of these sets would include all elements of \(W\), making this statement true.

III. \(T \cap W = S\):

This statement means that the intersection of sets \(T\) and \(W\) is equal to set \(S\). Given that \(T\) is a subset of \(W\), it's possible that the intersection of \(T\) and \(W\) includes more elements than just those in \(S\), so this statement may not be true in general.

Based on the analysis:

- Statement I is true.

- Statement II is true.

Therefore, the answer is I and II.