Repeated communication and Ramsey graphs

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.