Ramsey and 2-local Ramsey numbers for disjoint unions of cycles

We prove an upper bound for the Ramsey number of the disjoint union of odd cycles generalizing a result of Denley [The Ramsey numbers for disjoint unions of cycles, Discrete Math. 149 (1996) 31–44]. Moreover, we study the relation between the 2-local Ramsey number R2-loc(G)R2-loc(G) and the Ramsey number R(G)R(G), where G is a disjoint union of odd cycles. We give the exact value of the 2-local Ramsey number in the case when each cycle of G has an odd order.