Choiceless Ramsey Theory of Linear Orders

Abstract

Motivated by work of Erdős, Milner and Rado, we investigate symmetric and asymmetric partition relations for linear orders without the axiom of choice. The relations state the existence of a subset in one of finitely many given order types that is homogeneous for a given colouring of the finite subsets of a fixed size of a linear order. We mainly study the linear orders 〈^α2,< _{l e x} 〉, where α is an infinite ordinal and < _{l e x} is the lexicographical order. We first obtain the consistency of several partition relations that are incompatible with the axiom of choice. For instance we derive partition relations for 〈^ω2,< _{l e x} 〉 from the property of Baire for all subsets of ^ω2 and show that the relation $\langle ^{\kappa }{2}, <_{lex}\rangle \longrightarrow (\langle ^{\kappa }{2}, <_{lex}\rangle )^{2}_{2}$ is consistent for uncountable regular cardinals κ with κ ^<κ = κ. We then prove a series of negative partition relations with finite exponents for the linear orders 〈^α2,< _{l e x} 〉. We combine the positive and negative results to completely classify which of the partition relations $\langle ^{\omega }{2}, <_{lex}\rangle \longrightarrow (\bigvee _{\nu <\lambda }K_{\nu },\bigvee _{\nu <\mu }M_{\nu })^{m}$ for linear orders K _ν ,M _ν and m≤4 and 〈^ω2,< _{l e x} 〉→(K,M)ⁿ for linear orders K,M and natural numbers n are consistent.