Past exam of the graph Theory and Probabilistic Combinatorics course of the University of Oxford / 2011 / B / 5 / c

If there are no triangles, any edge colouring works. Otherwise $\Delta \ge 2$. Put $q=\lceil 3\sqrt{\Delta }\rceil$ and colour each edge independently and uniformly from $q$ colours. For each triangle $T$, let $A_T$ be the event that its three edges have one colour. ThenEvents for triangles sharing no edge depend on disjoint edge-colour variables, so each event is jointly independent of the collection of all such nonneighbours. Each edge of a triangle lies in at most $\Delta -2$ other triangles: a common neighbour of its endpoints must be one of at most $\Delta -1$ neighbours besides the other endpoint, and one of these is the third vertex of $T$ itself. Thus the dependency degree is at most $d=3(\Delta -2)$.

Because $q^2\ge 9\Delta$,The symmetric local lemma gives positive probability of avoiding every monochromatic triangle. HenceFor an edgeless graph the empty colouring handles even the zero-colour case $\Delta =0$.