A note on the rainbow Turán number of brooms with length 2 handles

For a fixed graph F, the rainbow Turán number ex^* (n, F) is the largest number of edges possible in an n-vertex graph which admits a rainbow-F-free proper edge-coloring. We focus on the rainbow Turán numbers of trees obtained by appending some number of pendant edges to one end of a length 2 path; we call such a tree with k total edges a k-edge broom with length 2 handle, denoted by B₊, ₂. Study of ex^* (n, B₊, ₂) was initiated by Johnston and Rombach, who claimed a proof asymptotically establishing the value of ex^* (n, B₊, ₂) for all k. We correct an error in this original argument, identifying two small cases in which the value claimed in the literature is incorrect; in all other cases, we recover the originally claimed value. Our argument also characterizes the extremal constructions for ex^* (n, B₊, ₂) for certain congruence classes of n modulo k.