Let H be a fixed finite graph and let → H be a hom-property, i.e. the set of all graphs admitting a homomorphism into H. We extend the definition of → H to include certain infinite graphs H and then describe the minimal reducible bounds for → H in the lattice of additive hereditary properties and in the lattice of hereditary properties.
For given graphs G₁,G₂,...,Gₖ, k ≥ 2, the multicolor Ramsey number R(G₁,G₂,...,Gₖ) is the smallest integer n such that if we arbitrarily color the edges of the complete graph on n vertices with k colors, then it is always a monochromatic copy of some , for 1 ≤ i ≤ k. We give a lower bound for k-color Ramsey number R(Cₘ,Cₘ,...,Cₘ), where m ≥ 8 is even and Cₘ is the cycle on m vertices. In addition, we provide exact values for Ramsey numbers R(P₃,Cₘ,Cₚ), where P₃ is the path on 3 vertices, and several...
We give the multicolor Ramsey number for some graphs with a path or a cycle in the given sequence, generalizing a results of Faudree and Schelp [4], and Dzido, Kubale and Piwakowski [2,3].