 
Summary: Repeated Communication and Ramsey Graphs
Noga Alon
Alon Orlitsky
Abstract
We study the savings afforded by repeated use in two zeroerror communication problems. We
show that for some random sources, communicating one instance requires arbitrarilymany bits,
but communicating multiple instances requires roughly one bit per instance. We also exhibit
sources where the number of bits required for a single instance is comparable to the source's size,
but two instances require only a logarithmic number of additional bits. We relate this problem
to that of communicating information over a channel. Known results imply that some channels
can communicate exponentially more bits in two uses than they can in one use.
1 Introduction
Starting with graph definitions below, this section introduces the two coding problems, describes
the results obtained, and relates them to known ones. The proofs are given in Sections 2 and 3.
Section 4 outlines possible extensions.
A graph G consists of a set V of vertices and a collection E of edges, unordered pairs of distinct
vertices. If {x, x } E, we say that x and x are connected in G. When E needs not be mentioned
explicitly, we write {x, x } G. An independent set in G is a collection of its vertices, no two
connected. G's independence number, (G), is the size of its largest independent set. A coloring
of G is an assignment of colors to its vertices such that connected vertices are assigned different
