Submission date: 27 April 2023

Abstract:

We consider bipartite graphs definable in o-minimal structures, in which the edge relation G is a finite union of graphs of certain measure-preserving maps.
We establish a fact on the existence of definable matchings with few short augmenting paths. Under the additional assumptions that G ⊆ [0,1]^n and 2-regularity, this yields the existence of definable matchings covering all vertices outside of a set of arbitrarily small positive measure (Lebesgue measure of the standard part). As an application we obtain an approximate 2-cancellation result for the semigroup of definable subsets of [0,1]^n modulo an equivalence relation induced by measure-preserving maps.

Mathematics Subject Classification: 03C64

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Full text arXiv 2304.14527: pdf, ps.