. Y}-?-s), 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))

S. Let-s-e-=-{s-?-x-\-{e}-:-s-=-s-?-x-for-some-s-?-s}, =. {s-?-x-\-{e}, :. , S. Belong-to, S. Then et al., 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

R. Ahlswede and N. Cai, A counterexample to Kleitman's conjecture concerning an edge-isoperimetric problem, Combinatorics, Probability and Computing, vol.8, pp.301-305, 1999.

H. Bandelt and V. Chepoi, Metric graph theory and geometry: a survey, Surveys on Discrete and Computational Geometry, vol.453, pp.49-86, 2008.

S. Bezrukov, Edge isoperimetric problems on graphs, Proc. Bolyai Math. Studies, vol.449, 1998.

N. Bousquet and S. Thomassé, VC-dimension and Erdös-Pósa property, Discr. Math, vol.338, pp.2302-2317, 2015.

A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance Regular Graphs, 1989.

N. Cesa-bianchi and D. Haussler, A graph-theoretic generalization of Sauer-Shelah lemma, Discr. Appl. Math, vol.86, pp.27-35, 1998.

V. Chepoi, Basis graphs of even Delta-matroids, J. Combin. Th. Ser B, vol.97, pp.175-192, 2007.

V. Chepoi, Distance-preserving subgraphs of Johnson graphs, Combinatorica

V. Chepoi, A. Labourel, and S. Ratel, On density of subgraphs of Cartesian products

M. Deza and M. Laurent, Geometry of Cuts and Metrics, 1997.

V. Diego, O. Serra, and L. Vena, On a problem by Shapozenko on Johnson graphs, 2016.

R. Diestel, Graduate texts in mathematics, Graph Theory, 1997.

D. ?. Djokovi?, Distance-preserving subgraphs of hypercubes, J. Combin. Th. Ser. B, vol.14, pp.263-267, 1973.

M. R. Garey and R. L. Graham, On cubical graphs, J. Combin. Th. B, vol.18, pp.84-95, 1975.

S. Hanneke and L. Yang, Minimax analysis of active learning, J. Mach. Learn. Res, vol.16, pp.3487-3602, 2015.

L. H. Harper, Optimal assignments of numbers to vertices, SIAM J. Appl. Math, vol.12, pp.131-135, 1964.

L. H. Harper, Global Methods for Combinatorial Isoperimetric Problems, Cambridge Studies in Advanced Mathematics, issue.90, 2004.

D. Haussler, Sphere packing numbers for subsets of the Boolean n-cube with bounded Vapnik-Chervonenkis dimension, J. Comb. Th. Ser. A, vol.69, pp.217-232, 1995.

D. Haussler, N. Littlestone, and M. K. Warmuth, Predicting {0, 1}-functions on randomly drawn points, vol.115, pp.248-292, 1994.

D. Haussler and P. M. Long, A generalization of Sauer's lemma, J. Combin. Th. Ser. A, vol.71, pp.219-240, 1995.

D. Kuzmin and M. K. Warmuth, Unlabelled compression schemes for maximum classes, J. Mach. Learn. Res, vol.8, pp.2047-2081, 2007.

S. B. Maurer, Matroid basis graphs I, J. Combin. Th. Ser. B, vol.14, pp.216-240, 1973.

B. K. Natarajan, On learning sets and functions, Machine Learning, vol.4, pp.67-97, 1989.

D. Pollard, Convergence of Stochastic Processes, 2012.

B. I. Rubinstein, P. L. Bartlett, and J. H. Rubinstein, Shifting: one-inclusion mistake bounds and sample compression, J. Comput. Syst. Sci, vol.75, pp.37-59, 2009.

N. Sauer, On the density of families of sets, J. Combin. Th., Ser. A, vol.13, pp.145-147, 1972.

S. V. Shpectorov, On scale embeddings of graphs into hypercubes, Europ. J. Combin, vol.14, pp.117-130, 1993.

V. N. Vapnik and A. Y. Chervonenkis, On the uniform convergence of relative frequencies of events to their probabilities, Theory Probab. Appl, vol.16, pp.264-280, 1971.