Ramsey numbers r(K3, G) for connected graphs G of order seven

AbstractThe triangle-graph Ramsey numbers are determined for all 814 of the 853 connected graphs of order seven. For the remaining 39 graphs lower and upper bounds are improved

An exploration in Ramsey theory

We present several introductory results in the realm of Ramsey Theory, a subfield of Combinatorics and Graph Theory. The proofs in this thesis revolve around identifying substructure amidst chaos. After showing the existence of Ramsey numbers of two types, we exhibit how these two numbers are related. Shifting our focus to one of the Ramsey number types, we provide an argument that establishes the exact Ramsey number for h(k, 3) for k ≥ 3; this result is the highlight of this thesis. We conclude with facts that begin to establish lower bounds on these types of Ramsey numbers for graphs requiring more substructure

Lower bounds for multi-colored Ramsey numbers from group orbits

In this paper the algorithm developed in [RK] for 2-color Ramsey numbers is generalized to multi-colored Ramsey numbers. All the cyclic graphs yielding the lower bounds R(3,3,4)\u3e=30, R(3,3,5)\u3e=45, and R(3,4,4)\u3e=55 were obtained. The two last bounds are apparently new