When performing a d-shifting ? ij with respect to a pair {e i , e j } such that {e i , e j } ? {e, y} = ?, a set S ? S can be mapped to a set ? ij (S) belonging to the fiber F ({e, y}) ,

, Also we have not found a counterexample to the following question (where the square of the clique-VC-dimension or of the star-VC-dimension is replaced by the product of the classical VC-dimension of S and the clique number of G, vol.1

, Is it true that for any set family S with 1,2-inclusion graph G 1,2 (S) = (V, E), d = VC-dim(S), and clique number ? = ?(G 1,2 (S))

As above, the bottleneck in solving Question 2 via shifting is that this operation may increase the clique number of 1,2-inclusion graphs. An alternative approach to Questions 1 and 2 is to adapt the original proof of Theorem 2 given in [19]. In brief, for a set family S of VC-dimension d and an element e, VC-dim(S e ) ? d, and VC-dim(S e ) ? d ? 1 hold. Denote by G e and G e the 1-inclusion graphs of S e and S e . Then |E(G e )| ? d|V (G e )| = d|S e | and E(G e ) ? (d ? 1)|V (G e )| = (d ? 1)|S e | by induction hypothesis. The proof of the required density inequality follows by induction from the equality |V (G)| = |S| = |S e | + |S e | = |V (G e )| + |V (G e )| and the inequality |E(G)| ? |E(G e )| + |E(G e )| + |V (G e )|. Unfortunately, as was the case for shiftings, vol.1 ,

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