Summary: Induced subgraphs with distinct sizes
April 1, 2008
We show that for every 0 < < 1/2, there is an n0 = n0( ) such that if n > n0 then
every n-vertex graph G of size at least n
2 and at most (1 - ) n
2 contains induced
k-vertex subgraphs with at least 10-7k different sizes, for every k n
This is best possible, up to a constant factor. This is also a step towards a conjecture
by Erdos, Faudree and S´os on the number of distinct pairs (|V (H)|, |E(H)|) of induced
subgraphs of Ramsey graphs.
AMS Subject Classification: 05C35, 05D40
Keywords: Induced subgraphs, size of subgraphs
For a graph G = (V, E), let hom(G) denote the maximum number of vertices in a clique
or an independent set in G. An n-vertex graph is c-Ramsey, if hom(G) c log n. Erdos,
Faudree and S´os (see , ) raised the following conjecture.