A Methodologically Pure Proof of a Convex Geometry Problem

We prove, using the minimalist axiom system for convex geometry proposed by W. A. Coppel, that, given n red and n blue points, such that no three are collinear, one can pair each of the red points with a blue point such that the n segments which have these paired points as endpoints are disjoint. MSC 2000: 51G05, 52A01, 52B11, 51K05 The problem stated in the abstract appeared as problem A-4 on the 1979 North American W. L. Putnam Competition. There the 2n points are considered to be in the standard Euclidean plane. As stated by us, as a theorem of Coppel’s minimalist convex geometry, this problem is a statement that describes a universal property of any reasonable geometric order relation, which we believe is general and interesting enough to justify a study of its possible proofs. Here is the only proof I know to exist in print (in both [8] and [5]): “There is a finite number (actually n!) of ways of pairing each of the red points with a blue point in a 1-to-1 way. Hence there exists a pairing for which the sum of the lengths of the segments joining paired points is minimal.” It is then shown that for such a pairing no two of the n segments intersect. There is a major discrepancy between the statement of this problem and its proof. In the statement there appear only notions of betweenness and collinearity, whereas the proof uses the notion of length. This goes against the principle of purity of the method of proof, enunciated by Hilbert a century ago: “In modern mathematics [. . .] one strives to preserve the purity of the method, i. e. to use in 0138-4821/93 $ 2.50 c © 2001 Heldermann Verlag 402 V. Pambuccian: A Methodologically Pure Proof of a Convex Geometry Problem the proof of a theorem as far as possible only those auxiliary means that are required by the content of the theorem.” ([7, p. 27]) As contributions to this programme, of providing methodologically pure proofs of important geometry problems we cite H. S. M. Coxeter’s proof from axioms of betweenness and incidence of the Sylvester-Gallai problem ([3], [4]), and the proofs from affine axioms of a generalized Euler line theorem by A. W. Boon [1] and E. Snapper [12]. The proof that we shall present, which respects the Hilbertian purity requirement, proceeds inside the following axiomatic framework: There is one kind of individual variables, points, and a ternary relation among points B of betweenness (with B(abc) to be read as ‘b lies between a and c (b could be equal to a or to c)’). When we say ‘segment ab intersects segment cd’ we mean ‘there is a point x, different from a, b, c, d such that B(axb) and B(cxd)’, and when we say that ‘x, y and z are collinear’ (for which we use the abbreviation L(xyz)) we mean B(xyz)∨B(yzx)∨B(zxy). The axioms are: