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Abstract.A Borel subset of the unit square whose vertical and horizontal sections are two-point sets admits a natural group action. We exploit this to discuss some questions about Borel subsets of the unit square on which every function is a sum of functions of the coordinates. Connection with probability measures with prescribed marginals and some function algebra questions is discussed.Introduction. Given non-empty sets X and Y , a subset of X × Y is called a matching if it is the graph of a one-to-one map of X onto Y . The question as to which subsets S of X × Y contain a matching has been of combinatorial and set-theoretic interest [12]. By a theorem of D. König [11], if each section S x = {y ∈ Y : (x, y) ∈ S}, x ∈ X, and S y = {x ∈ X : (x, y) ∈ S}, y ∈ Y , consists of exactly n elements (n finite; for infinite n see

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