With the notions of configurations and subconfigurations in hand, we are now ready to define monotone properties andmonotone parameters.

Definitions

We have already discussed the monotonicity of the happy ending problem.

We may formalize a property either as a subset of the set of all configurations or as the characteristic function of this subset, a function of configurations that is true for the configurations with the property and false for the ones without it.

We can also formproperties fromparameters.

Many classical problems in the discrete geometry of point sets can be formulated in terms of monotone properties and monotone parameters of configurations.

Obstacles

Every monotone property of configurations can be characterized by its obstacles, the smallest configurations that do not have the property.

A puzzle from the Russian mathematical olympiads asks for a proof that, for any convex pentagon of points in a grid, there is another grid point on or inside the smaller convex pentagon formed by the diagonals of the given pentagon (Figure 12.1). Thus, the grid has no empty pentagons. The existence or nonexistence of empty polygons is not a monotone property, but it suggests the study of other properties, beyond the basic ones studied in Chapter 11, based on what the convex polygons of a configuration contain and not just which points formtheir vertices.

Weak Convexity

A configuration S is in weakly convex position if none of the points is interior to the convex hull of S: each point either is a vertex of the convex hull (as in configurations that are in convex position) or lies on a convex hull edge. For instance, grid(2, n) is weakly convex for every n, but grid(3, 3) is not. It is tempting to guess that the weakly convex configurations are defined by the property FORBIDDEN(TETRAD), by analogy to the fact that convex = FORBIDDEN (TETRAD, LINE(3)). The three-point line is allowed in weakly convex positions, so it should be dropped fromthe list of obstacles.However, this guess is not quite right. There is one configuration that is not in weakly convex position but has no tetrad: the quincunx (Figure 12.2), a configuration formed from the four points of a convex quadrilateral together with one more point where the diagonals of the quadrilateral cross.

The two obstacles LINE(3) and TETRAD for convex position were derived from Carathéodory's theorem, that every point in the convex hull of a set of points belongs to the convex hull of two or three points from the set. The corresponding result for weakly convex position is Steinitz's theorem that every point interior to the convex hull is interior to the convex hull of three or four points from the set. Therefore, the two forbidden patterns for weakly convex position are the tetrad and the quincunx.

We begin our study with an examination of the different ways we can place small numbers of points in the plane, and what it means for two sets of points to be different.

Small Configurations

In how many different ways can we place n points in the plane? With a list of all of the possible placements, we could prove statements such as Klein's observation about convex quadrilaterals in five-point sets, automatically, merely by checking all the cases. When n is small enough, we can provide an explicit answer.

So far, we have used only an intuitive notion for what it means for two sets of points to be the same or different. But when we get to five points, we already need to define more precisely what we mean. Figure 3.2 depicts 13 sets of five points. Are they all different from each other? Two of these are mirror images: should that count as two different sets of points or as two different views of a single way of placing five points?

We will give a more precise definition of what it means for sets of points to be the same or different in the next section. For the definition we use, mirror images are not (in general) considered to be the same, so there are indeed 13 different ways of arranging five points. Open Problem 3.5, later in this chapter, formalizes the problem of counting configurations of different sizes.

The numbers of sets of n points grow very quickly, as cn log n for a constant c > 1. Aichholzer et al. (2002) used computer searches to establish a database of small sets of points. To help control the size of the database, they exclude sets that have three points on a line and count mirror-image sets of points as equivalent. Despite these restrictions, the sets in their database still grow as cn log n , but with a smaller c.

A planar graph is a graph that can be drawn in the plane with its vertices as points and its edges as noncrossing curves (often drawn as line segments).Universal point sets are point sets that can form the vertex set of any sufficiently small planar graph. Before defining this concept more carefully as the monotone parameter universal (Definition 16.1), we look at a computer puzzle that has popularized some of these concepts.

Planarity

Planarity is a computer puzzle game based on planar graph drawing, originally devised by John Tantalo and Mary Radcliffe. In it, one is presented with a tangled drawing of a graph (Figure 16.1). The goal is to untangle it, by moving the vertices one by one, until the result has no crossings. As the vertices move, the edges move with them as straight line segments. Each level of the puzzle presents a more complicated graph to be untangled. Although there exist computer algorithms that can solve these puzzles efficiently, their visual complexity nonetheless makes them a challenge to human puzzle-solvers.

The original version of this puzzle generates its graphs from arrangements of randomly generated lines (Figure 16.2, left). However, the sharp angles of these arrangements make it difficult to reconstruct them while solving the puzzle. Indeed, even finding an arrangement that generates a given graph is a hard problem, much harder than finding an untangled drawing for the graph.

Therefore, it can be easier to search for a solution to the puzzle that places the vertices on a grid (or a rough human approximation to a grid), such as the one in Figure 16.2, right. But when we're just starting the puzzle, we don't know what the whole grid drawing will look like, if it even exists. How big should we make the grid squares? If they're too big, the drawing won't fit in the window, and we'll have to waste time and moves shrinking it. But if they're too small, it will be more difficult to precisely place each of the graph vertices into its grid position.

In the early 1930s, Hungarian mathematician Esther Klein made a discovery that, despite its apparent simplicity, would kick off two major lines of research in mathematics. Klein observed that every set of five points in the plane has either three points in a line or four points in a convex quadrilateral. This became one of the first results in the two fields of discrete geometry (the study of combinatorial properties of geometric objects such as points in the Euclidean plane, and the subject of this book) and Ramsey theory (the study of the phenomenon that unstructured mathematical systems often contain highly structured subsystems).

Klein's observation can be proven by a simple case analysis that considers how many of the points belong to their convex hull. The convex hull is a convex polygon, having some of the given points as its vertices and containing the others. It can be defined mathematically in many ways, for instance as the smallest-area convex polygon that contains all of the given points or as the largest-area simple polygon whose vertices all belong to the given points. The convex hull of points that are not all on a line always has at least three vertices (for otherwise it could not enclose a nonzero area) and, for five given points, at most five vertices. If it has five vertices, any four of them forma convex quadrilateral, and if it has four vertices then it is a convex quadrilateral. The remaining possibility for the convex hull is a triangle, with the other two points either part of a line of three points or inside the triangle.When both points are inside, and the line through themmisses the triangle vertices, it alsomisses one side of the triangle. In this case the two interior points and the two points on the missed side forma convex quadrilateral (Figure 1.1).

The challenge of extending and generalizing this observation was taken up by two of Klein's friends, Paul Erdʺos and George Szekeres. They proved that, for every k, a convex k-gon can be found in all large enough sets of points, as long as no three of the points lie on a line. Klein later married Szekeres, and their marriage is commemorated in the name of Erdʺos and Szekeres's result: the happy ending theorem.