Families of nested graphs with compatible symmetric-group actions

Abstract

For fixed positive integers n and k, the Kneser graph $$KG_{n,k}$$ has vertices labeled by k-element subsets of $$\{1,2,\dots ,n\}$$ and edges between disjoint sets. Keeping k fixed and allowing n to grow, one obtains a family of nested graphs, each of which is acted on by a symmetric group in a way which is compatible with these inclusions and the inclusions of each symmetric group into the next. In this paper, we provide a framework for studying families of this kind using the $${{\,\mathrm{FI}\,}}$$-module theory of Church et al. (Duke Math J 164(9):1833–1910, 2015), and show that this theory has a variety of asymptotic consequences for such families of graphs. These consequences span a range of topics including enumeration, concerning counting occurrences of subgraphs, topology, concerning Hom-complexes and configuration spaces of the graphs, and algebra, concerning the changing behaviors in the graph spectra.