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Section 1.5 Combinatorics and Geometry

There are many problems in geometry that are innately combinatorial or for which combinatorial techniques shed light on the problem.

Example 1.12.

In Figure 1.13, we show a family of \(4\) lines in the plane. Each pair of lines intersects and no point in the plane belongs to more than two lines. These lines determine \(11\) regions.

Figure 1.13. Lines and regions

Under these same restrictions, how many regions would a family of \(8947\) lines determine? Can different arrangements of lines determine different numbers of regions?

Example 1.14.

Mandy says she has found a set of \(882\) points in the plane that determine exactly \(752\) lines. Tobias disputes her claim. Who is right?

Example 1.15.

There are many different ways to draw a graph in the plane. Some drawings may have crossing edges while others don't. But sometimes, crossing edges must appear in any drawing. Consider the graph \(G\) shown in Figure 1.16.

Figure 1.16. A graph with crossing edges

Can you redraw \(G\) without crossing edges?

Suppose Sam and Deborah were given a homework problem asking whether a particular graph on \(2843952\) vertices and \(9748032\) edges could be drawn without edge crossings. Deborah just looked at the number of vertices and the number of edges and said that the answer is “no.” Sam questions how she can be so certain—without looking more closely at the structure of the graph. Is there a way for Deborah to justify her definitive response?