It is well known that $H_n(X)=0$ when $X$ is a CW complex of dimension smaller than $n$. But I wonder if it is still true when we replace $X$ with a CW pair $(X,A)$.
When I try to prove this, the result doesn't seem to follow directly from the long exact sequence for $(X,A)$.
The result in fact does not follow from the long exact sequence for a pair, in the following sense: it is possible to write down spaces $A \subseteq X$ such that $A$ and $X$ have homological dimension less than $n$ (meaning their homology vanishes in degree $n$ above) but $X/A$ has homological dimension $n$. A simple example is to take $X = D^2, A = S^1$, which have homological dimensions $0$ and $1$ respectively, but $X/A \cong S^2$ has homological dimension $2$.
You can instead do it using the fact that $X/A$ is also a CW complex of dimension less than $n$, as Pedro says in the comments.