Another Abstraction of the Erdös-Szekeres Happy End Theorem

The Happy End Theorem of Erdős and Szekeres asserts that for every integer n greater than two there is an integer N such that every set of N points in general position in the plane includes the n vertices of a convex n-gon. We generalize this theorem in the framework of certain simple structures, which we call “happy end spaces”. In the winter of 1932/33, Esther Klein observed that from any set of five points in the plane of which no three lie on the same line it is always possible to select four points that are vertices of a convex polygon. When she shared this news with a circle of her friends in Budapest, the following prospect of generalizing it emerged: Can we find for each integer n greater than two an integer N(n) such that from any set of N(n) points in the plane of which no three lie on the same line it is always possible to select n points that are vertices of a convex polygon? Endre Makai proved that N(5) = 9 works here. A few weeks later, George Szekeres proved the existence of N(n) for all n. His argument produced very large upper bounds for N(n): for instance, it gave N(5) 6 2. Soon afterwards, Paul Erdős came up with a different proof, which led to much smaller values of N(n): Sackler School of Mathematics and Blavatnik School of Computer Sciences, Tel Aviv University, Tel Aviv, Israel Department of Computer Science and Software Engineering, Concordia University, Montréal, Québec, Canada Department of Computer Science and Software Engineering, Concordia University, Montréal, Québec, Canada 5264 av. Henri-Julien app. 3, Montréal, Québec, Canada the electronic journal of combinatorics 17 (2010), #N11 1 (⋆) From any set of ( 2n−4 n−2 ) +1 points in the plane of which no three lie on the same line it is always possible to select n points that are vertices of a convex polygon. In December 1934, Erdős and Szekeres submitted for publication a manuscript containing both proofs; the paper [6] appeared in 1935. Esther Klein and George Szekeres got married on June 13, 1937 and Paul Erdős began referring to (⋆) as The Happy End Theorem. Abstractions of this theorem have been studied by Korte and Lovász [7] and by Morris and Soltan [8]. These deal with abstract convexity spaces satisfying the anti-exchange property and the simplex partition property, and having a finite Caratheodory number c > 3. See [8], [7] for the precise definitions and more details. A somewhat less abstract version is considered in [10], based on the order type of a configuration of points in the plane. This enables the authors to show, using an exhaustive computer search, that any configuration of 17 points in general position in the plane contains a convex 6-gon. Here we propose another abstraction: A happy-end space is a set S along with a function f : S × S × S → {+,−}ions of this theorem have been studied by Korte and Lovász [7] and by Morris and Soltan [8]. These deal with abstract convexity spaces satisfying the anti-exchange property and the simplex partition property, and having a finite Caratheodory number c > 3. See [8], [7] for the precise definitions and more details. A somewhat less abstract version is considered in [10], based on the order type of a configuration of points in the plane. This enables the authors to show, using an exhaustive computer search, that any configuration of 17 points in general position in the plane contains a convex 6-gon. Here we propose another abstraction: A happy-end space is a set S along with a function f : S × S × S → {+,−} such that f(x, y, z) = f(y, x, z) for all x, y, z; in this space, a subset C of S is called convex if, and only if, for every subset B of C such that |B| > 2 and for every point x of B, there is another y in B such that f(x, y, z) is constant on B − {x, y}. (In this definition, the only values of f(x, y, z) that matter are those where x, y, z are all distinct.) Every set S of points in the plane such that no three lie on the same line and no two have the same first coordinate defines a happy-end space by f(x, y, z) = { +1 if point z lies above the line xy, −1 if point z lies below the line xy; in this space, a set is convex if and only if it consists of vertices of a convex polygon. In this abstract setting, the Happy End Theorem generalizes as follows: Theorem 1. For every positive integer n there is a positive integer N such that every happy-end space on N points contains a convex set of n points. Proof. Following Erdős and Rado [4, 5], we let a → (b)r denote the statement that whenever the k-point subsets of an a-point set are coloured by r colours, there is a b-point set whose k-point subsets are all of the same colour. Frank Ramsey [9] proved that for every choice of positive integers b, k, r, there is an integer a such that a → (b)r . the electronic journal of combinatorics 17 (2010), #N11 2 We claim that if N satisfies N → (n)8 then it also satisfies the conclusion of Theorem 1. To justify this claim, consider an arbitrary happy-end space on N points, impose a linear order ≺ on its underlying set S and, for each set T of three points in S, write g(T ) = (f(v, w, u), f(u, w, v), f(u, v, w)), where T = {u, v, w} and u ≺ v ≺ w. Since N → (n)8, there are a set C of n points in S and a vector (s1, s2, s3) in {+1,−1}3 such that g(T ) = (s1, s2, s3) for every three-point subset T of C; this means that