Two results about the hypercube

First we consider families in the hypercube Q_n with bounded VC dimension. Frankl raised the problem of estimating the number m(n,k) of maximum families of VC dimension k. Alon, Moran and Yehudayoff showed that n^{(1+o(1))[Formula presented][Formula presented]}≤m(n,k)≤n^{(1+o(1))[Formula presented]}.We close the gap by showing that logm(n,k)=(1+o(1))[Formula presented]logn. We also show how a bound on the number of induced matchings between two adjacent small layers of Q_n follows as a corollary. Next, we consider the integrity I(Q_n) of the hypercube, defined as I(Q_n)=min{|S|+m(Q_n∖S):S⊆V(Q_n)},where m(H) denotes the number of vertices in the largest connected component of H. Beineke, Goddard, Hamburger, Kleitman, Lipman and Pippert showed that c[Formula presented]≤I(Q_n)≤C[Formula presented]logn and suspected that their upper bound is the right value. We prove that the truth lies below the upper bound by showing that I(Q_n)≤C[Formula presented]logn.