The zero-error side information problem and chromatic numbers (Corresp.)

A discrete random variable
X
is to be transmitted by means of a discrete signal. The receiver has prior knowledge of a discrete random variable
Y
jointly distributed with
X
. The probability of error must be exactly zero, and the problem is to minimize the signal's alphabet size. In the case where the transmitter also has access to the value of
Y
, the problem is trivial and no advantage can be obtained by block coding over independent repetitions. If, however,
Y
is not known at the transmitter then the problem is equivalent to the chromatic number problem for graphs, and block coding may produce savings.