Large monochromatic components of small diameter

Gyarfas conjectured in 2011 that every r-edge-colored K n contains a monochromatic component of bounded (perhaps three) diameter on at least n / ( r - 1 ) vertices. Letzter proved this conjecture with diameter four. In this note we improve the result in the case of r = 3: We show that in every 3-edge-coloring of K n either there is a monochromatic component of diameter at most three on at least n / 2 vertices or every color class is spanning and has diameter at most four.

Automated summary

In a 3-edge-coloring of Kn, there either exists a monochromatic component with a diameter of at most three spanning at least n / 2 vertices, or every color class is spanning and has a diameter of at most four.