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...

For two graph H and G, the Ramsey number r(H,G) is the smallest positive integer n such that every red-blue edge coloring of the complete graph Kn on n vertices contains either a red copy of H or a blue copy of G. Motivated by questions of Erdős and Harary, in this note we study how the Ramsey number r(Ks, G) depends on the size of the graph G. For s ≥ 3, we prove that for every G with m edges,...

A graph G is r-Ramsey for a graph H, denoted by G→ (H)r, if every r-colouring of the edges of G contains a monochromatic copy of H. The graph G is called r-Ramseyminimal for H if it is r-Ramsey for H but no proper subgraph of G possesses this property. Let sr(H) denote the smallest minimum degree of G over all graphs G that are r-Ramseyminimal for H. The study of the parameter s2 was initiated ...

A subgraph of an edge-coloured graph is rainbow if all of its edges have different colours. For graphs G and H the anti-Ramsey number ar(G,H) is the maximum number of colours in an edge-colouring of G with no rainbow copy of H. The notion was introduced by Erdős, Simonovits and V. Sós and studied in case G = Kn. Afterwards exact values or bounds for anti-Ramsey numbers ar(Kn, H) were establishe...

The Ramsey number for a graph G versus a graph H, denoted by R(G, H), is the smallest positive integer n such that for any graph F of order n, either F contains G as a subgraph or F contains H as a subgraph. In this paper, we investigate the Ramsey numbers for union of trees versus small cycle and small wheel. We show that if ni is odd and 2ni+1 ≥ ni for every i, then R( Sk i=1 Tni , W4) = R(Tn...

A (finite) sequence of nonnegative integers is graphic if it is the degree sequence of some simple graph G. Given graphs G1 and G2, we consider the smallest integer n such that for every n-term graphic sequence π, there is some graph G with degree sequence π with G1 ⊆ G or with G2 ⊆ G. When the phrase “some graph” in the prior sentence is replaced with “all graphs” the smallest such integer n i...

The Ramsey number for a graph G versus a graph H, denoted by R(G,H), is the smallest positive integer n such that for any graph F of order n, either F contains G as a subgraph or F contains H as a subgraph. In this paper, we investigate the Ramsey numbers for union of stars versus small cycle and small wheel. We show that if ni ≥ 3 for i = 1, 2, . . . , k and ni ≥ ni+1 ≥ √ ni − 2, then R( ∪k i=...

We determine the 2-color Ramsey number of a connected triangle matching c(nK3) which is a graph with n vertex disjoint triangles plus (at least n − 1) further edges that keep these triangles connected. We obtain that R(c(nK3), c(nK3)) = 7n − 2, somewhat larger than in the classical result of Burr, Erdős and Spencer for a triangle matching, R(nK3, nK3) = 5n. The motivation is to determine the Ra...

We write H → G if every 2-coloring of the edges of graph H contains a monochromatic copy of graph G. A graph H is G-minimal if H → G, but for every proper subgraph H ′ of H, H ′ 6→ G. We define s(G) to be the minimum s such that there exists a G-minimal graph with a vertex of degree s. We prove that s(Kk) = (k − 1) and s(Ka,b) = 2 min(a, b) − 1. We also pose several related open problems. c © 2...

A color pattern is a graph whose edges are partitioned into color classes. A family F of color patterns is a Ramsey family if there is some integer N such that every edgecoloring of KN has a copy of some pattern in F . The smallest such N is the (pattern) Ramsey number R(F ) of F . The classical Canonical Ramsey Theorem of Erdős and Rado [4] yields an easy characterization of the Ramsey familie...