An extremal characterization of projective planes

In this article, we prove that amongst all n by n bipartite graphs of girth at least six, where n = q(2) + q + 1 >= 157, the incidence graph of a projective plane of order q, when it exists, has the maximum number of cycles of length eight. This characterizes projective planes as the partial planes with the maximum number of quadrilaterals

On the classical Ramsey Number R(3,3,3,3)

The classical Ramsey Number R(3, 3, 3, 3), which is the smallest positive integer n such that any edge coloring with four colors of the complete graph on n vertices must contain at least one monochromatic triangle, is discussed. Basic facts and graph theoretic definitions are given. Papers concerning triangle-free colorings are presented in a historical overview. Mathematical theory underlying the main result of the thesis, which is Richard Kramers unpublished result, i?(3,3,3,3) \u3c 62, is given. The algorithms for the com putational verification of this result are presented along with a discussion of the software tools that were utilized to obtain it