Suppose we are given a convex n-gon and all possible pairs of non adjacent vertices are joined to form chords. Let f(n) be the number of pairs of intersecting chords. So f ( 4 ) = 1 , f ( 5 ) = 5. I want to determine f(n) in general.Some thoughts:Its not clear to me as to why f(n) is well defined. f ( n ) = f ( n − 1 ) + something. I want to determine that something.Is f(n) an upper bound for the crossing number of K n ?

Question

Suppose we are given a convex n-gon and all possible pairs of non adjacent vertices are joined to form chords. Let f(n) be the number of pairs of intersecting chords. So   f  (  4  )  =  1  ,  f  (  5  )  =  5. I want to determine f(n) in general.Some thoughts:Its not clear to me as to why f(n) is well defined.  f  (  n  )  =  f  (  n  −  1  )  + something. I want to determine that something.Is f(n) an upper bound for the crossing number of       K    n  ?

yorbakid2477w6 · Accepted Answer

Step 1We need to assume the vertices are in &quot;general position.&quot;Step 2Take any 4 vertices. One pair of chords determined by these vertices meets inside the polygon, the other two do not. So the number of chord intersections inside the polygon is                     (                    n      4                      )            .

Josiah Owens · Accepted Answer

Step 1If you want to approach it using your method of   f  (  n  )  =  f  (  n  −  1  )  + something, then observe that when we introduce the additional vertex V, any new point of intersection must involve this vertex V.Step 2Any point of intersection will have to involve 3 vertices from the other   n  −  1. It is clear that given any set of 3 vertices, there is a unique point of intersection, obtained by adjoining V to the &#039;middle&#039; vertex. Hence &quot;something&quot;   =                              (                                      n          −          1                3                              )                    .Step 3You can now solve the recurrence to show that   f  (  n  )  =                              (                            n        4                              )                    .

Suppose we are given a convex n-gon and all possible pairs of non adjacent vertices are joined to form chords. Let f(n) be the number of pairs of intersecting chords. So f(4)=1,f(5)=5. I want to determine f(n) in general.

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