Using computational techniques we derive six new upper bounds on the classical twocolor Ramsey numbers: R(3, 10) ≤ 42, R(3, 11) ≤ 50, R(3, 13) ≤ 68, R(3, 14) ≤ 77, R(3, 15) ≤ 87, and R(3, 16) ≤ 98. All of them are improvements by one over the previously best published bounds. Let e(3, k, n) denote the minimum number of edges in any triangle-free graph on n vertices without independent sets of o...

Ramsey number R(m,n) = r is the smallest integer r such that a graph of r vertices has either a complete subgraph (clique) of size m or its complement has a complete subgraph of size n. Currently, the exact value of R(5,5) is unknown, however the best known lower and upper bounds are 43 <= R(5,5) <= 49. In this paper we will discuss a method that we use to construct a better lower bound, namely...

Simplicial versions of topological abelian gauge theories are constructed which reproduce the continuum expressions for the partition function and Wilson expectation value of linked loops, expressible in terms of R–torsion and linking numbers respectively. The new feature which makes this possible is the introduction of simplicial fields (cochains) associated with the dual triangulation of the ...

graph is called a complete r-partite graph if its vertex set can be partitioned into r independent sets V1, . . . , Vr such that for any i = 1, 2, . . . , r every vertex in Vi is adjacent to all other vertices in Vj , j 6= i. We denote a complete r-partite graph with part sizes |Vi| = ni by Kn1,...,nr . For a graph G and a positive integer k we use kG to denote k vertex-disjoint copies of G. Gi...

In this article we give the generalized triangle Ramsey numbers R(K3, G) of 12 005 158 of the 12 005 168 graphs of order 10. There are 10 graphs remaining for which we could not determine the Ramsey number. Most likely these graphs need approaches focusing on each individual graph in order to determine their triangle Ramsey number. The results were obtained by combining new computational and th...

We define the (q, r)-Whitney numbers of the first and second kinds in terms of the q-Boson operators, and obtain several fundamental properties such as recurrence formulas, orthogonality and inverse relations, and other interesting identities. As a special case, we obtain a q-analogue of the r-Stirling numbers of the first and second kinds. Finally, we define the (q, r)-Dowling polynomials in t...

The Ramsey number R(C4,Km) is the smallest n such that any graph on n vertices contains a cycle of length four or an independent set of order m. With the help of computer algorithms we obtain the exact values of the Ramsey numbers R(C4,K9) = 30 and R(C4,K10) = 36. New bounds for the next two open cases are also presented.

Let the numbers P (r, n, k) be defined by P (r, n, k) := Pr ( H n −H (1) k , · · · , H (r) n −H (r) k ) , where Pr(x1, · · · , xr) = (−1)Yr(−0!x1,−1!x2, · · · ,−(r− 1)!xr) and Yr are the exponential complete Bell polynomials. By observing that the numbers P (r, n, k) generate two Riordan arrays, we establish several general summation formulas, from which series of harmonic number identities are...