On a Ramsey-type problem

Suppose a graph G has the property that if one colors the edges of G in r colors, there always exists a monochromatic triangle. Is it true that if one colors the edges of G in r + 1 colors so that every vertex is incident to at most r colors then there must be a monochromatic triangle? This problem, which was first raised by P. Erdös, is answered in the negative here.