  We investigate the effect of the well-known Mycielski construction on the
Shannon capacity of graphs and on one of its most prominent upper bounds, the
(complementary) Lov\'asz theta number. We prove that if the Shannon capacity of
a graph, the distinguishability graph of a noisy channel, is attained by some
finite power, then its Mycielskian has strictly larger Shannon capacity than
the graph itself. For the complementary Lov\'asz theta function we show that
its value on the Mycielskian of a graph is completely determined by its value
on the original graph, a phenomenon similar to the one discovered for the
fractional chromatic number by Larsen, Propp and Ullman. We also consider the
possibility of generalizing our results on the Sperner capacity of directed
graphs and on the generalized Mycielsky construction. Possible connections with
what Zuiddam calls the asymptotic spectrum of graphs are discussed as well.
