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In this note we develop and clarify some of the basic combinatorial properties of the new notion of n-dependence (for 1 ≤ n < ω) recently introduced by Shelah [She07]. In the same way as dependence of a theory means its inability to encode a bipartite random graph with a definable edge relation, n-dependence corresponds to the inability to encode a random (n + 1)-partite (n + 1)-hypergraph with a definable edge relation. Most importantly, we characterize n-dependence by counting ϕ-types over finite sets (generalizing Sauer-Shelah lemma and answering a question of Shelah from [She05]) and in terms of the collapse of random ordered (n + 1)-hypergraph indiscernibles down to order-indiscernibles (which implies that the failure of n-dependence is always witnessed by a formula in a single free variable).

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The β‐hydrogen‐free imidazole monomer, 2‐methyl‐N‐methyl‐4(5)‐vinylimidazole (2M4VIm), is synthesized to prepare anion exchange membranes (AEMs). The radiation‐induced graft polymerization of 2M4VIm and styrene is performed in poly(ethylene‐co‐tetrafluoroethylene), followed by N‐alkylation and ion exchange reactions to prepare 4‐vinylimidazolium‐containing AEM (2M4VIm/St‐AEM). AEMs that have a 6/4 2M4VIm/St molar ratio and 1.7 mmol g−1 ion exchange capacity (IEC) result in 75 mS cm−1 conductivity and 60% water uptake at room temperature in water, demonstrating enhanced values compared to previously reported 2‐methyl‐N‐vinylimidazolium‐containing AEMs (2MNVIm/St‐AEM) that have a similar comonomer ratio and IEC. The small‐angle neutron scattering measurements of dry and swollen 2M4VIm/St‐AEM reveal hydrophilic/hydrophobic two‐phase separation, as observed for 2MNVIm/St‐AEM with a similar Im/St composition. 2M4VIm/St‐AEMs exhibit higher alkaline stability in 1 m KOH at 80 °C at an early stage owing to the suppression of imidazolium β‐elimination but lower long‐term stability than that of 2MNVIm/St‐AEM due to the ring‐opening reaction of the imidazolium group due to its outward orientation from the graft main chain.

We study model companions of theories extending the graph axioms. First we prove general results concerning the existence of the model companion. Then, by applying these results to the case of graphs, we give a series of companionable and non-companionable examples.