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...

This paper introduces a general methodology, based on abstraction and symmetry, that applies to solve hard graph edge-coloring problems and demonstrates its use to provide further evidence that the Ramsey number R(4, 3, 3) = 30. The number R(4, 3, 3) is often presented as the unknown Ramsey number with the best chances of being found “soon”. Yet, its precise value has remained unknown for more ...

Given a graph H and an integer k≥1, the Gallai–Ramsey number GRk(H) is defined to be minimum n such that every k-edge coloring of complete Kn contains either rainbow (all different colored) triangle or monochromatic copy H. In this paper, we study numbers for graphs with chromatic three as K̂m m≥2, where kipas m+1 vertices obtained from join K1 Pm, class five vertices, denoted by ℋ. We first gen...

Given a graph H and an integer r ≥ 2, let G → (H, r) denote the Ramsey property of a graph G, that is, every r-coloring of the edges of G results in a monochromatic copy of H. Further, let m(G) = maxF⊆G |E(F )|/|V (F )| and define the Ramsey density minf (H, r) as the infimum of m(G) over all graphs G such that G → (H, r). In the first part of this paper we show that when H is a complete graph ...

Let Q(n, c) denote the minimum clique number over graphs with n vertices and chromatic number c. We investigate the asymptotics of Q(n, c) when n/c is held constant. We show that when n/c is an integer α, Q(n, c) has the same growth order as the inverse function of the Ramsey number R(α+ 1, t) (as a function of t). Furthermore, we show that if certain asymptotic properties of the Ramsey numbers...