MathDB

Give a point set on a plane

P=\{P_1,P_2,\cdots,P_{1994}\}

. Any three points in

P

are not colinear. Divide points in

P

into

83

groups, in each group there are at least three points, and any point exactly belongs to one group. Connect any two points in the same group with a line segment, but we do not connect points not in the same group. Now we get a figure

G

. Note the number of triangles in

G

m(G)

. (a) Find the minumum value of

m(G)

m_0

. (b)

G*

is one of the figures that

m(G)=m_0

, color the line segments in

G*

with four colors. Prove that there exists a proper way, satisfying that no triangle is made up of three sides in the same color.

Problem Statement