  We apply the Ramsey theory to the analysis of geometrical properties of
closed contours. Consider a set of six points placed on a closed contour. The
straight lines connecting these points are y_ik (x)={\alpha}_ik x+\b{eta}_ik
(i,k=1...6), {\alpha}_ik is not equal to 0. We color the edges connecting the
points for which {\alpha}_ik>0 holds with red, and the edges for which
{\alpha}_ik<0 with green with red. At least one monochromic triangle should
necessarily appear within the curve (according to the Ramsey number R(3,3)=6).
This result is immediately applicable for the analysis of dynamical billiards.
The second theorem emerges from the combination of the Jordan curve and Ramsey
theorem. The closed curve is considered. We connect the points located within
the same region with green links and the points placed within the different
regions with red links. In this case, transitivity/intransitivity of the
relations between the points should be considered. Ramsey constructions arising
from the differential geometry of closed contours are discussed.
