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We compute the model-theoretic Grothendieck ring, \(K_0({\mathcal {Q}})\), of a dense linear order (DLO) with or without end points, \({\mathcal {Q}}=(Q,u003c)\), as a structure of the signature \(\{u003c\}\), and show that it is a quotient of the polynomial ring over \({\mathbb {Z}}\) generated by \({\mathbb {N}}_+\times (Q\sqcup \{-\infty \})\) by an ideal that encodes multiplicative relations of pairs of generators. This ring can be embedded in the polynomial ring over \({\mathbb {Q}}\) generated by \(Q\sqcup \{-\infty \}\). As a corollary we obtain that a DLO satisfies the pigeon hole principle for definable subsets and definable bijections between them—a property that is too strong for many structures.

Definable combinatorics with dense linear orders