Splittings and Ramsey Properties of Permutation Classes

We say that a permutation π is merged from permutations ρ and τ , if we can color the elements of π red and blue so that the red elements are order-isomorphic to ρ and the blue ones to τ . A permutation class is a set of permutations closed under taking subpermutations. A permutation class C is splittable if it has two proper subclasses A and B such that every element of C can be obtained by merging an element of A with an element of B. Several recent papers use splittability as a tool in deriving enumerative results for specific permutation classes. The goal of this paper is to study splittability systematically. As our main results, we show that if σ is a sum-decomposable permutation of order at least four, then the class Av(σ) of all σ-avoiding permutations is splittable, while if σ is a simple permutation, then Av(σ) is unsplittable. We also show that there is a close connection between splittings of certain permutation classes and colorings of circle graphs of bounded clique size. Indeed, our splittability results can be interpreted as a generalization of a theorem of Gyárfás stating that circle graphs of bounded clique size have bounded chromatic number.