Graphs Having Small Number of Sizes on Induced k-Subgraphs

@article{Axenovich2007GraphsHS,
  title={Graphs Having Small Number of Sizes on Induced k-Subgraphs},
  author={Maria Axenovich and J{\'o}zsef Balogh},
  journal={SIAM J. Discrete Math.},
  year={2007},
  volume={21},
  pages={264-272}
}
Let ` be any positive integer, n be a sufficiently large number, and let G be a graph on n vertices. Define for any k νk(G) = |{|E(H)| : H is an induced subgraph of G on k vertices}|. We show that if 2` ≤ k ≤ n−2` and νk(G) ≤ ` then G has a complete or an empty subgraph on at least n − ` + 1 vertices, and a homogeneous set of order at least n − 2` + 2. These results are sharp. 

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