Small Ramsey Numbers

We present data which, to the best of our knowledge, includes all known nontrivial values and bounds for specific graph, hypergraph and multicolor Ramsey numbers, where the avoided graphs are complete or complete without one edge. Many results pertaining to other more studied cases are also presented. We give references to all cited bounds and values, as well as to previous similar compilations. We do not attempt complete coverage of asymptotic behavior of Ramsey numbers, but concentrate on their specific values

Computers in Ramsey Theory; Testing, Constructions and Nonexistence

Computers in Ramsey Theory Ramsey theory is often regarded as the study of how order emerges from randomness. While originated in mathematical logic, it has applications in geometry, number theory, game theory, information theory, approximation algorithms, and other areas of mathematics and theoretical computer science. Ramsey theory studies the conditions of when a combinatorial object necessarily contains some smaller given objects. The central concept in Ramsey theory is that of arrowing, which in the case of graphs describes when colorings of larger graphs necessarily contain monochromatic copies of given smaller graphs. The role of Ramsey numbers is to quantify some of the general existential theorems in Ramsey theory, always involving arrowing. The determination of whether this arrowing holds is notoriously difficult, and thus it leads to numerous computational challenges concerning various types of Ramsey numbers and closely related Folkman numbers. This talk will overview how computers are increasingly used to study the bounds on Ramsey and Folkman numbers, and properties of Ramsey arrowing in general. This is happening in the area where traditional approaches typically call for classical computer-free proofs. It is evident that now we understand Ramsey theory much better than a few decades ago, increasingly due to computations. Further, more such progress and new insights based on computations should be anticipated

The Ramsey numbers R(K_3, K_8 - e) and R(K_3, K_9 - e)

We give a general construction of a triangle free graph on 4p points whose complement does not contain K_p+2 - e for p \u3e= 4. This implies the the Ramsey number R(K_3, K_k - e) \u3e= 4k - 7 for k \u3e= 6. We also present a cyclic triangle free graph on 30 points whose complement does not contain K_9 - e. The first construction gives lower bounds equal to the exact values of the corresponding Ramsey number for k = 6, 7 and 8. the upper bounds are obtained by using computer algorithms. In particular, we obtain two new values of Ramsey numbers R(K_3, K_8 - e) = 25 and R(K_3, K_9 - e) = 31, the bounds 36 \u3c= R(K_3, K_10 - e) \u3c= 39, and the uniqueness of extremal graphs for Ramsey numbers R(K_3, K_6 - e) and R(K_3, K_7 - e)

Enumeration of all simple t-(t+7,t+1,2) designs

We enumerate by computer algorithms all simple t (t +7, t +1, 2) designs for 1 \u3c= t \u3c= 5, i.e. for all possible t , and this enumeration is new for t \u3e= 3. The number of nonisomorphic designs is equal to 3, 13, 27, 1 and 1 for t = 1, 2, 3, 4 and 5, respectively. We also present some properties of these designs including orders of their full automorphism groups and resolvability

On the Ramsey number R(K_5 - e,K_5 - e)

Using computer algorithms we found that there exists a unique, up to isomorphism, graph on 21 points and 125 graphs on 20 points for the Ramsey number R(K_5 - e, K_5 - e) = 22. We also construct all graphs on n points for the Ramsey number R(K_4 - e, K_ 5 - e) = 13 for all n \u3c= 12

Some Computational and Theoretical Problems for Ramsey Numbers

We discuss some computational challenges and related open questions concerning classical Ramsey numbers. This talk overviews known constructive bounds for the difference between consecutive Ramsey numbers and presents what is known about the most studied cases including $R(5,5)$ and $R(3,3,4)$. Although the main problems we discuss are concerned with concrete cases, and they involve significant computational approaches, there are interesting and important theoretical questions behind each of them

Paths, cycles and wheels in graphs without antitriangles

We investigate paths, cycles and wheels in graphs with independence number of at most 2, in particular we prove theorems characterizing all such graphs which are hamiltonian. Ramsey numbers of the form R (G,K3), for G being a path, a cycle or a wheel, are known to be 2n (G) - 1, except for some small cases. In this paper we derive and count all critical graphs 1 for these Ramsey numbers

Finding Simple T-designs by Using Basis Reduction

In 1976, Kramer and Mesner observed that finding a t-design with a given automorphism group can be reduced to solving a matrix problem of the form AX=M, X[i]=0 or I, for all i, I \u3c =i=6 and duplicate the results of Leavitt, Kramer and Magliveras [3,10] in substantially shorter time. Furthermore, a new simple 6-design was found using the algorithm described in this pape

Upper bounds for some Ramsey numbers R(3, k)

Using several computer algorithms we calculate some values and bounds for the function e(3, k, n), the minimum number of edges in a triangle-free graphs on n vertices with no independent set of size k. As a consequence, the following new upper bounds for the classical two color Ramsey numbers are obtained: R(3,10)\u3c=43, R(3,11)\u3c=51, R(3,12)\u3c=60, R(3,13)\u3c=69 and R(3,14)\u3c=78

Towards deciding the existance of 2-(22,8,4) designs

We report on progress on towards deciding the existence of 2-(22,8,4) designs without assuming any automorphisms. Using computer algorithms we have shown that in any such design every two blocks have nonempty intersection, every quadruple of points can occur in at most two blocks, and no three blocks can pairwise intersect in one point