Ramsey numbers R(K3,G) for graphs of order 10

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 theoretical results. We also describe an optimized algorithm for the generation of all maximal triangle-free graphs and triangle Ramsey graphs. All Ramsey numbers up to 30 were computed by our implementation of this algorithm. We also prove some theoretical results that are applied to determine several triangle Ramsey numbers larger than 30. As not only the number of graphs is increasing very fast, but also the difficulty to determine Ramsey numbers, we consider it very likely that the table of all triangle Ramsey numbers for graphs of order 10 is the last complete table that can possibly be determined for a very long time.Comment: 24 pages, submitted for publication; added some comment

The Study of Ramsey numbers r(C_k, C_k, C_k)

The Ramsey number r(C_k, C_k, C_k), denoted as r_3(C_k), is the smallest positive integer n such that any edge coloring with three colors of the complete graph on n vertices must contain at least one monochromatic cycle C_k. In this project, most literature on the Ramsey numbers r_3(C_k) are overviewed. Algorithms to check if a graph G contains any specific path or cycle and to construct extremal graphs for cycle C_k are developed. All good 3-colorings of complete graph K_10 are constructed to verify the value of Ramsey number r_3(C_4). Ramsey number value of r_3(C_3) is verified by direct point by point extension algorithm. The lower bounds for the Ramsey numbers r_3(C_5), r_3(C_6), and r_3(C_7) are provided as well. Additionally, the possibility of further research for larger k, especially for r_3(C_8) and r_3(C_10) is searched. Most of the results are based on computer algorithms

Computation of the Ramsey Numbers R(C_4, K_9) and R(C_4, K_10)

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

Ramsey numbers R(K3, G) for graphs of order 10

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 theoretical results. We also describe an optimized algorithm for the generation of all maximal triangle-free graphs and triangle Ramsey graphs. All Ramsey numbers up to 30 were computed by our implementation of this algorithm. We also prove some theoretical results that are applied to determine several triangle Ramsey numbers larger than 30. As not only the number of graphs is increasing very fast, but also the difficulty to determine Ramsey numbers, we consider it very likely that the table of all triangle Ramsey numbers for graphs of order 10 is the last complete table that can possibly be determined for a very long time

Bounds onSome Ramsey Numbers Involving Quadrilateral

For graphs G_1,G_2,...,G_m the Ramsey number R(G_1,G_2,...,G_m) is defined to be the smallest integer n such that any m-coloring of the edges of the complete graph K_n must include a monochromatic G_i in color i, for some i. In this note we establish several lower and upper bounds for some Ramsey numbers involving quadrilateral C_4, including R(C_4,K_9) \u3c = 32, 19 \u3c = R(C_4,C_4,K_4) \u3c = 22, 31 \u3c = R(C_4,C_4,C_4,K_4) \u3c = 50, 52 \u3c = R(C_4,K_4,K_4) \u3c = 72, 42 \u3c = R(C_4,C_4,K_3,K_4) \u3c = 76, and 87 \u3c = (C_4,C_4,K_4,K_4) \u3c = 179

Semidefinite Programming and Ramsey Numbers

We use the theory of flag algebras to find new upper bounds for several small graph and hypergraph Ramsey numbers. In particular, we prove the exact values R(K−, K−, K−) = 28, R(K8, C5) = 29, R(K9, C6) = 41, R(Q3, Q3) = 13, R(K3,5, K1,6) = 17, R(C3, C5, C5) = 17, and R(K−, K−; 3) = 12, and in addition improve many additional upper bounds