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We investigate the effect of the well-known Mycielski construction on theShannon 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 ofa graph, the distinguishability graph of a noisy channel, is attained by somefinite power, then its Mycielskian has strictly larger Shannon capacity thanthe graph itself. For the complementary Lov\'asz theta function we show thatits value on the Mycielskian of a graph is completely determined by its valueon the original graph, a phenomenon similar to the one discovered for thefractional chromatic number by Larsen, Propp and Ullman. We also consider thepossibility of generalizing our results on the Sperner capacity of directedgraphs and on the generalized Mycielsky construction. Possible connections withwhat Zuiddam calls the asymptotic spectrum of graphs are discussed as well.

Shannon capacity, Lov\'asz theta number and the Mycielski construction