For a scientific paper, I need to write these sentences as one sentence using "respectively".
Arnold and Benson  proved the claim when X is a rainbow set.
Carlson  proved the claim when X is a cloudy set.
Arnold and Davison  proved the claim when X is a rainy set.
My first attempt:
When X is a rainbow set, a cloudy set, and a rainy set, the claim was proved by Arnold and Benson , Carlson , and Arnold and Davison , respectively.
However, I am worried this might be ambiguous because there are more people than types of sets and the additional "and"s complicate the list. (Note: Arnold is the co-author of two papers.)
Is my attempt sound, and are there better ways to write this?