The ith Ramsey number for matchings is determined. In addition, our results lead to the calculation of the Ramsey index for matchings.
The purpose of this paper is to calculate the ith Ramsey number for matchings. In order to state our results, we will need some notation. Any undefined notation follows Behzad,. For i, n e Z ÷, we write K(i; n) instead of K (i,..., i) to denote the complete n-partite graph whose parts each contain i vertices. In particular, K(1, n) is the usual star graph while K(1; n) is the complete graph Kn. For k e Z ÷ and graphs G, G1,..., Gk, we write
is an arbitrary (edge) factorization of G, then Gj c Fj for some j.
For i, k e Z ÷ and graphs G1,..., G~, the ith Ramsey number ri(G~,..., Gk) is defined to be the least positive integer p such that K(i;p)--> (G1, ..., Gk). In particular, r~(G1,..., Gk) is the generalized Ramsey number r(G1,..., Gk). Moreover, if i <~j and K(i;p)--> (G1,. • •, Gk), then K(i;p) c K(j;p) and therefore K(j;p)--> (G1,..., G~). So ri(G1,..., Gk)~>r~(G1,..., Gk) whenever i<<-j and therefore ri(G1,..., Gk)<~r(G1,..., Gk) for each i e Z ÷. It follows that ith Ramsey numbers do indeed exist. In addition, the sequence {ri(G~,..., Gk)}~=l is a non-increasing sequence of positive integers and therefore convergent. The Ramsey index i(G~, ..., Gk) is defined to be the least positive integer I such that rt(G1,..., Gk)= limi__,~ ri(G~,..., Gk). Obviously, ri(K2,. • •, K2) = 2 for each i and therefore i(K2,..., K2) = 1.
The above "Ramsey numbers" were introduced by Benedict [2]. In addition, Benedict extended the work of Burr and Roberts [5] by calculating the ith Ramsey number and Ramsey index for stars. In a further attempt to extend currently known ith Ramsey numbers, we calculate the ith Ramsey number and Ramsey index for matchings. We require the following theorem.
Theorem 1 (Cockayne and Lorimer [6]). If k ~ Z + and ma,..., mk ~ Z + with ml >-mj for each j, then r(mlK2, . . . , mkK2) = ml + 1 + •k=l (mj --1). Now we present some preliminary results before establishing the main