Duality between combinatorial Banach spaces holds precisely when the families are the finite cliques and anti-cliques of a perfect graph on the naturals, making Lovász' perfect graph theorem a corollary, with further study of the Sierpiński graph case.
SIERPI ´NSKI,Sur un problème de la théorie des relations, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, 2 (1933), pp
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Geometric duality, perfect graphs, and the Sierpi\'nski space
Duality between combinatorial Banach spaces holds precisely when the families are the finite cliques and anti-cliques of a perfect graph on the naturals, making Lovász' perfect graph theorem a corollary, with further study of the Sierpiński graph case.