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.