The goal of the paper is to develop some of the ideas of N. J. Kalton and N. L. Randrianarivony [Math. Ann. 341, No. 1, 223–237 (2008; Zbl 1146.46050)] in order to be able to apply them to the case of quasi-reflexive Banach spaces.
Let \(\mathbb{M}\) be an infinite subset of \(\mathbb{N}\), denote by \((G_k(\mathbb{M}),d)\) the metric space introduced by Kalton and Randrianarivony [loc. cit.]. The authors introduce the set \(I_k(\mathbb{M})\) of the strictly interlaced pairs of \(G_k(\mathbb{M})\) by \[ I_k(\mathbb{M})=\{(n_1,\dots,n_k,m_1,\dots, m_k) \in G_k(\mathbb{M})\times G_k(\mathbb{M}),\ n_1 < m_1 < n_2 < m_2 < \dots < n_k < m_k\}. \] One of the main results is the following (Theorem 2.2): Let \(p\in (1,\infty)\) and \(Y\) be a quasi-reflexive \(p\)-asymptotically uniformly smooth Banach space. Then there exists a constant \(C>0\) such that, for any infinite subset \(\mathbb{M}\) of \(\mathbb{N}\), any Lipschitz function \(f:(G_k(\mathbb{M}),d) \to Y^{**}\), and any \(\varepsilon>0\), there exists an infinite subset \(\mathbb{M}'\) of \(\mathbb{M}\) such that \[ \forall (n_1,\dots,n_k,m_1,\dots, m_k) \in I_k(\mathbb{M}') :\ \ \ \|f(n_1,\dots,n_k) - f(m_1,\dots, m_k)\| \leq C \text{Lip}(f)k^{\frac{1}{p}} + \varepsilon. \] This result is used to find relations between embeddability and the Banach-Saks properties (previous results of this type were obtained by B. M. Braga [Stud. Math. 237, No. 1, 71–97 (2017; Zbl 1380.46013)]). Out of the results of this type, we state only Proposition 3.2, whose statement does not require technical definitions: Assume that \(X\) is a Banach space which coarse Lipschitz embeds into a quasi-reflexive asymptotically uniformly smooth Banach space \(Y\). Then \(X\) has the alternating Banach-Saks property.