In this work we consider the diversity maximization problem, where given a data set $X$ of $n$ elements, and a parameter $k$, the goal is to pick a subset of $X$ of size $k$ maximizing a certain diversity measure. [CH01] defined a variety of diversity measures based on pairwise distances between the points. A constant factor approximation algorithm was known for all those diversity measures except ``remote-matching'', where only an $O(\log k)$ approximation was known. In this work we present an $O(1)$ approximation for this remaining notion. Further, we consider these notions from the perpective of composable coresets. [IMMM14] provided composable coresets with a constant factor approximation for all but ``remote-pseudoforest'' and ``remote-matching'', which again they only obtained a $O(\log k)$ approximation. Here we also close the gap up to constants and present a constant factor composable coreset algorithm for these two notions. For remote-matching, our coreset has size only $O(k)$, and for remote-pseudoforest, our coreset has size $O(k^{1+\varepsilon})$ for any $\varepsilon > 0$, for an $O(1/\varepsilon)$-approximate coreset.