On the ramsey numbers N(3,3,…3;2)

AbstractThe main results of this paper are N(3,3,3,3;2) > 50 and f(k+1)≥3 f(k)+f(k−2), where f(k) = N3,3,…;2)ktimes −1 for k ≥ 3

An algorithmic approach for multi-color Ramsey graphs

The classical Ramsey number R(r1,r2,...,rm) is defined to be the smallest integer n such that no matter how the edges of Kn are colored with the m colors, 1, 2, 3, . . . ,m, there exists some color i such that there is a complete subgraph of size ri, all of whose edges are of color i. The problem of determining Ramsey numbers is known to be very difficult and is usually split into two problems, finding upper and lower bounds. Lower bounds can be obtained by the construction of a, so called, Ramsey graph. There are many different methods to construct Ramsey graphs that establish lower bounds. In this thesis mathematical and computational methods are exploited to construct Ramsey graphs. It was shown that the problem of checking that a graph coloring gives a Ramsey graph is NP-complete. Hence it is almost impossible to find a polynomial time algorithm to construct Ramsey graphs by searching and checking. Consequently, a method such as backtracking with good pruning techniques should be used. Algebraic methods were developed to enable such a backtrack search to be feasible when symmetry is assumed. With the algorithm developed in this thesis, two new lower bounds were established: R(3,3,5)≥45 and R(3,4,4)≥55. Other best known lower bounds were matched, such as R(3,3,4)≥30. The Ramsey graphs giving these lower bounds were analyzed and their full symmetry groups were determined. In particular it was shown that there are unique cyclic graphs up to isomorphism giving R(3,3,4)≥30 and R(3,4,4)≥55, and 13 non-isomorphic cyclic graphs giving R(3,3,5)≥45

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 Mathematical theory underlying the main result of the thesis, which is Richard Kramers unpublished result, i?(3,3,3,3) \u3c 62, is given.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