Edge-coloring to avoid large monochromatic stars

Conjecture (proposed by Erdős, Faudree, Rousseau and Schelp [1])

In every \(3\)-coloring of the edges of \(K_n\), there is a color such that there are three vertices whose neighbors (joining by edges in such a color) include at least two-thirds of all the vertices.

An example of Kierstead [1] shows the result of the conjecture is the best possible. In 2014, the conjecture was proved by Baber and Talbot [2], using flag algebras.

1 R. Faudree, C. C. Rousseau and R. H. Schelp, Problems in graph theory from Memphis, The Mathematics of Paul Erdős, II (R. L. Graham and J. Nešetril, eds.), 7-26, Springer-Verlag, Berlin, 1996.
2 R. Baber and J. Talbot, A solution to the 2/3 conjecture, SIAM J. Discrete Math, 28(2), 756-766, 2014.