  Standard mixed-integer programming formulations for the stable set problem on
$n$-node graphs require $n$ integer variables. We prove that this is almost
optimal: We give a family of $n$-node graphs for which every polynomial-size
MIP formulation requires $\Omega(n/\log^2 n)$ integer variables. By a
polyhedral reduction we obtain an analogous result for $n$-item knapsack
problems. In both cases, this improves the previously known bounds of
$\Omega(\sqrt{n}/\log n)$ by Cevallos, Weltge & Zenklusen (SODA 2018).
  To this end, we show that there exists a family of $n$-node graphs whose
stable set polytopes satisfy the following: any $(1+\varepsilon/n)$-approximate
extended formulation for these polytopes, for some constant $\varepsilon > 0$,
has size $2^{\Omega(n/\log n)}$. Our proof extends and simplifies the
information-theoretic methods due to G\"o\"os, Jain & Watson (FOCS 2016, SIAM
J. Comput. 2018) who showed the same result for the case of exact extended
formulations (i.e. $\varepsilon = 0$).
