Degree multiplicities and independent sets in K4-free graphs
Erdos et al.  asked for what values of k there is a sequence of graphs (Gn)∞n=1 such that Gn has n vertices and independence number o(n), contains no K4, and no k + 1 vertices have the same degree. In  it is proved that such a k has to be at least 4, but the problem whether any such k exists is left open. By making use of some graphs constructed by Bollobás and Erdos , we shall prove that for k = 5 (and so for k ≥ 5) there is such a sequence (Gn)∞n=1.
Bollobás, B. (1996). Degree multiplicities and independent sets in K4-free graphs. Discrete Mathematics, 158 (1-3), 27-35. https://doi.org/10.1016/0012-365X(95)00002-E