# On density of subgraphs of halved cubes

Let S be a family of subsets of a set X of cardinality m and VC-dim( S) be the Vapnik-Chervonenkis dimension of S. Haussler, Littlestone, and Warmuth (Inf. Comput., 1994) proved that if G_1( S)=(V,E) is the subgraph of the hypercube Q_m induced by S (called the 1-inclusion graph of S), then |E|/|V|<VC-dim( S). Haussler (J. Combin. Th. A, 1995) presented an elegant proof of this inequality using the shifting operation. In this note, we adapt the shifting technique to prove that if S is an arbitrary set family and G_1,2( S)=(V,E) is the 1,2-inclusion graph of S (i.e., the subgraph of the square Q^2_m of the hypercube Q_m induced by S), then |E|/|V|<d2, where d:=cVC-dim^*( S) is the clique-VC-dimension of S (which we introduce in this paper). The 1,2-inclusion graphs are exactly the subgraphs of halved cubes and comprise subgraphs of Johnson graphs as a subclass.

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