MathDB

A triangulation

\mathcal{T}

of a polygon

P

is a finite collection of triangles whose union is

P,

and such that the intersection of any two triangles is either empty, or a shared vertex, or a shared side. Moreover, each side of

P

is a side of exactly one triangle in

\mathcal{T}.

Say that

\mathcal{T}

is admissible if every internal vertex is shared by

6

or more triangles. For example [asy] size(100); dot(dir(-100)^^dir(230)^^dir(160)^^dir(100)^^dir(50)^^dir(5)^^dir(-55)); draw(dir(-100)--dir(230)--dir(160)--dir(100)--dir(50)--dir(5)--dir(-55)--cycle); pair A = (0,-0.25); dot(A); draw(A--dir(-100)^^A--dir(230)^^A--dir(160)^^A--dir(100)^^A--dir(5)^^A--dir(-55)^^dir(5)--dir(100)); [/asy] Prove that there is an integer

M_n,

depending only on

n,

such that any admissible triangulation of a polygon

P

with

n

sides has at most

M_n

triangles.

6

Problems(2)