The star chromatic index of $K_{3,3}$ is six

The theorem asserts that for the complete bipartite graph $G=K_{3,3}$, whose vertices are the disjoint union of two parts each containing three elements and whose edges are exactly all pairs joining different parts, there exists an edge-coloring $c$ using the six colors $\mathrm{Fin}\,6$ such that: for every vertex $v$ and distinct neighbors $u,w$ of $v$, the colors assigned to $vu$ and $vw$ are different; there do not exist five pairwise distinct vertices $v_0,v_1,v_2,v_3,v_4$ with all four consecutive edges present and with the color of $v_0v_1$ equal to that of $v_2v_3$ and the color of $v_1v_2$ equal to that of $v_3v_4$; and there do not exist four pairwise distinct vertices $v_0,v_1,v_2,v_3$ with all four cycle edges $v_0v_1,v_1v_2,v_2v_3,v_3v_0$ present and with the colors of opposite edges $v_0v_1,v_2v_3$ equal and the colors of $v_1v_2,v_3v_0$ equal. In addition, there does not exist any edge-coloring of the same graph using the five colors $\mathrm{Fin}\,5$ that satisfies all three of these conditions. Here an edge-coloring assigns a color to every edge of $G$.