Lower Bounds for Lower Ramsey Numbers Ralph Faudree Ronald zy J. Gould Michael S. Jacobson Linda Lesniak z MEMPHIS STATE UNIVERSITY EMORY UNIVERSITY UNIVERSITY OF LOUISVILLE DREW UNIVERSITY ABSTRACT For any graph G, let i(G) and p(G) denote the smallest number of vertices in a maximal independent set and maximal clique, respectively. For positive integers rn and n, the lower Ramsey number s(rn,n) is the largest integer p so that every graph of order p has i(G) zyxw I rn or p(G) I n. In this paper we give several new lower bounds for s (rn,n) as well as determine precisely the values s(1,n). INTRODUCTION zyxwvu In zyxwvut [3], Mynhardt introduced the concept of lower Ramsey numbers, which stemmed from the original idea of Ramsey numbers. For any undefined terms, see Chartrand and Lesniak [2]. The zyxw independence (clique) number of a graph G, denoted p(C) (w(G)), is the largest number of vertices in a maximal independent set (complete subgraph or clique) of G. The Ramsey number, r(m, n), is the smallest integer p so that every graph of order p has P(G) 2 m or o(G) zyxwv 1 n. To introduce the lower Ramsey number, we de- fine the parameters zyxwvu i(G) and p(G) to be the order of the smallest maximal independent set and smallest maximal clique, respectively. The lower Ram- sey number s(m,n) is the largest integer p so that every graph G or order p has i(G) I zy m or p(G) I n. The parameter i(G) has previously been stud- ied as a bound for the domination number of a graph, and has been given the name independent domination number (see [l]). In [3,4] Mynhardt gives several results for this new Ramsey-type parameter, including a proof that these numbers do in fact exist and are Journal of Graph Theory, Vol. 14, No. 6, 723-730 (1990) 0 1990 John Wiley & Sons, Inc. CCC 0364-9024/90/060723-08$04.00