Repeated communication and Ramsey graphs

We study the savings afforded by repeated use in two zero-error communication problems. Channel coding: Proving a correspondence between Ramsey numbers and independence numbers of normal graph powers, we show that some channels can communicate exponentially more bits in two uses than they can in one use, and that this is essentially the largest possible increase. Using probabilistic constructions of self-complementary Ramsey graphs, we show that similar results hold even when the number of transmissible bits is large relative to the channel's size. Dual-source coding: Using probabilistic colorings of directed line graphs, we show that there are dual sources where communicating one instance requires arbitrarily many bits, yet communicating many instances requires at most two bits per instance. For dual sources where the number of bits required for a single instance is comparable to the source's size, we employ probabilistic constructions of self-complementary Ramsey graphs that are also Cayley graphs to show that conveying two instances may require only a logarithmic number of additional bits over that needed to convey one instance.