Let G be a permutation group on a set $\Omega $ of size t. We say that $\Lambda \subseteq \Omega $ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $\Lambda $ . We define the height of G to be the maximum size of an independent set, and we denote this quantity $\textrm{H}(G)$ . In this paper, we study $\textrm{H}(G)$ for the case when G is primitive. Our main result asserts that either $\textrm{H}(G)< 9\log t$ or else G is in a particular well-studied family (the primitive large–base groups). An immediate corollary of this result is a characterization of primitive permutation groups with large relational complexity, the latter quantity being a statistic introduced by Cherlin in his study of the model theory of permutation groups. We also study $\textrm{I}(G)$ , the maximum length of an irredundant base of G, in which case we prove that if G is primitive, then either $\textrm{I}(G)<7\log t$ or else, again, G is in a particular family (which includes the primitive large–base groups as well as some others).

To deepen our understanding of organizations’ heterogeneous responses to institutional demand, we develop a ‘relational complexity’ argument to highlight organizations’ diverse institutional linkages as another important source of practice variation. We argue that diverse relations between organizations and the institutional authority can filter distinct institutional pressures and expectations, shape organizational interpretations of environmental demands, and thus trigger heterogeneous organizational practices. We adopt this theoretical framework and distinguish two types of institutional linkages with the state to understand different adoption patterns in corporate social responsibility (CSR) in its early stage of diffusion in China. Based on a national survey dataset consisting of 1,268 firms, our analyses show that firms having a stronger bureaucratic linkage with the state tend to focus on more visible external-oriented CSR practices. In contrast, those firms forming a closer partnership with the state through political or semi-political associations are more likely to take more extensive adoptions by further developing internal CSR structures. This study enriches the institutional analyses by shifting our attention to the relational dynamics between organizations and institutional authority as a key source of practice variation. It also has important implications to the research and practices of CSR in emerging economies.