 
Summary: Induced subgraphs of prescribed size
Noga Alon
Michael Krivelevich
Benny Sudakov
Abstract
A subgraph of a graph G is called trivial if it is either a clique or an independent set. Let
q(G) denote the maximum number of vertices in a trivial subgraph of G. Motivated by an open
problem of Erdos and McKay we show that every graph G on n vertices for which q(G) C log n
contains an induced subgraph with exactly y edges, for every y between 0 and n(C)
. Our
methods enable us also to show that under much weaker assumption, i.e., q(G) n/14, G still
must contain an induced subgraph with exactly y edges, for every y between 0 and e(
log n)
.
1 Introduction
All graphs considered here are finite, undirected and simple. For a graph G = (V, E), let (G)
denote the independence number of G and let w(G) denote the maximum number of vertices of a
clique in G. Let q(G) = max{(G), w(G)} denote the maximum number of vertices in a trivial
induced subgraph of G. By Ramsey Theorem (see, e.g., [10]), q(G) (log n) for every graph G
