Monochromatic diameter-2 components in edge colorings of the complete graph

@article{Ruszink2021MonochromaticDC,
  title={Monochromatic diameter-2 components in edge colorings of the complete graph},
  author={Mikl{\'o}s Ruszink{\'o} and Lang Song and Daniel P. Szabo},
  journal={Involve, a Journal of Mathematics},
  year={2021}
}
Gyarfas conjectured that in every r -edge-coloring of the complete graph K n there is a monochromatic component on at least n ∕ ( r − 1 ) vertices which has diameter at most 3. We show that for r = 3 , 4 , 5 and 6 a diameter of 3 is best possible in this conjecture, constructing colorings where every monochromatic diameter-2 subgraph has strictly less than n ∕ ( r − 1 ) vertices. 
Large monochromatic components of small diameter
TLDR
This note improves the result in the case of r = 3 and shows that in every 3-edge-coloring of Kn 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 least four.

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