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Repeated Communication and Ramsey Graphs Alon Orlitsky

Summary: Repeated Communication and Ramsey Graphs
Noga Alon
Alon Orlitsky
We study the savings afforded by repeated use in two zero-error communication problems. We
show that for some random sources, communicating one instance requires arbitrarily-many 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


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


Collections: Mathematics