Determine the first unresolved case \(B_{10,3}\)

Determine the asymptotic rainbow Turán number \(\mathrm{ex}^*(n,B_{10,3})\), resolving whether it equals \(\frac{9}{2}n+O(1)\) or is attained asymptotically by a non-clique extremal construction.

Background

The authors prove that no proper edge-coloring of $K_{11}$ avoids a rainbow $B_{10,3}$. Consequently, the standard clique construction using disjoint copies of $K_{10}$ gives the lower bound $\frac{9}{2}n+O(1)$, but it is not known whether this is optimal. The remaining possibility is that a denser construction not based on cliques determines the extremal value.