Rainbow clique subdivisions

We show that for any integer $t \ge 2$, every properly edge colored $n$-vertex graph with average degree at least $(\log n)^{2+o(1)}$ contains a rainbow subdivision of a complete graph of size $t$. Note that this bound is within $(\log n)^{1+o(1)}$ factor of the lower bound. This also implies a result on the rainbow Tur\'{a}n number of cycles.