Gallai-Ramsey number conjecture for the three-triangle fan

Let $F_n$ be the fan graph with $n$ triangles, defined by

$$F_n=K_1+n\overline{K_2},$$

so that $F_3$ is the fan with three triangles, and let $gr_k(K_3;F_3)$ denote the $k$-color Gallai-Ramsey number of $F_3$ versus $K_3$. The authors' conjecture. For $k\geq 2$,

$$gr_k(K_3;F_3)=\begin{cases} 14\cdot 5^{\frac{k-2}{2}}-1, & \text{if }k\text{ is even};\\ 33\cdot 5^{\frac{k-3}{2}}, & \text{if }k\text{ is odd}. \end{cases}$$

This asserts that the lower bound in the odd-$k$ case of the preceding theorem is sharp, completing the exact determination of the Gallai-Ramsey numbers for $F_3$. The even-$k$ case and the odd cases $k=3,5$ are proved in the paper, while the general odd case is left open.