Repeated Communication and Ramsey Graphs Alon Orlitsky Summary: Repeated Communication and Ramsey Graphs Noga Alon Alon Orlitsky Abstract 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 Collections: Mathematics