Decomposition horizons: from graph sparsity to model-theoretic dividing lines

09/15/2022

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by   Samuel Braunfeld, et al.

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Let 𝒞 be a hereditary class of graphs. Assume that for every p there is a hereditary NIP class 𝒟_p with the property that the vertex set of every graph G∈𝒞 can be partitioned into N_p=N_p(G) parts in such a way that the union of any p parts induce a subgraph in 𝒟_p and log N_p(G)∈ o(log |G|). We prove that 𝒞 is (monadically) NIP. Similarly, if every 𝒟_p is stable, then 𝒞 is (monadically) stable. Results of this type lead to the definition of decomposition horizons as closure operators. We establish some of their basic properties and provide several further examples of decomposition horizons.