Inclusion-exclusion polynomials with large coefficients

We prove that for every positive integer $k$ there exist an inclusion-exclusion polynomial $Q_{\{q_1,q_2,...,q_k\}}$ with the height at least $c^{2^k}\prod_{j=1}^{k-2}q_j^{2^{k-j-1}-1}$, where $c$ is a positive constant and $q_1<q_2<...<q_k$ are pairwise coprime and arbitrary large.

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