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Sizes of induced subgraphs of Ramsey graphs , Jozsef Balogh

Summary: Sizes of induced subgraphs of Ramsey graphs
Noga Alon
, J´ozsef Balogh
, Alexandr Kostochka
and Wojciech Samotij§
January 27, 2008
An n-vertex graph G is c-Ramsey if it contains neither a complete nor an empty
induced subgraph of size greater than c log n. Erdos, Faudree and S´os conjectured that
every c-Ramsey graph with n vertices contains (n5/2) induced subgraphs any two of
which differ either in the number of vertices or in the number of edges, i.e. the number
of distinct pairs (|V (H)|, |E(H)|), as H ranges over all induced subgraphs of G, is
(n5/2). We prove an (n2.3693) lower bound.
1 Introduction
For a graph G = (V, E), call a set W V homogenous, if W induces a clique or an
independent set. Let hom(G) denote the maximum size of a homogenous set of vertices of
G. For a positive constant c > 0, an n-vertex graph G is called c-Ramsey if hom(G) c log n.
Ramsey theory states that every n-vertex graph G satisfies hom(G) (log n)/2, and for
almost all such G, we have hom(G) 2 log n. In other words, in a random graph G, the
value hom(G) is of logarithmic order. Moreover, the only known examples of graphs with


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics