tailieunhanh - Báo cáo toán học: "New Lower Bound Formulas for Multicolored Ramsey Numbers"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: New Lower Bound Formulas for Multicolored Ramsey Numbers. | New Lower Bound Formulas for Multicolored Ramsey Numbers Aaron Robertson Department of Mathematics Colgate University Hamilton NY 13346 aaron@ Submitted July 26 2001 Accepted March 18 2002 MR Subject Classification 05D10 Abstract We give two lower bound formulas for multicolored Ramsey numbers. These formulas improve the bounds for several small multicolored Ramsey numbers. 1. INTRODUCTION In this short article we give two new lower bound formulas for edgewise r-colored Ramsey numbers R k-1 k2 . kr r 3 defined below. Both formulas are derived via construction. We will make use of the following notation. Let G be a graph V G the set of vertices of G and E G the set of edges of G. An r-coloring X will be assumed to be an edgewise coloring . x G E G 1 2 . rg. If u v 2 V G we take x u v to be the color of the edge connecting u and v in G. We denote by Kn the complete graph on n vertices. Definition Let r 2. Let ki 2 1 i r. The number R R k1 k2 . kr is defined to be the minimal integer such that any edgewise r-coloring of KR must contain for some j 1 j r a monochromatic Kkj of color j. If we are considering the diagonal Ramsey numbers . k1 k2 kr k we will use Rr k to denote the corresponding Ramsey number. The numbers R k1 k2 . kr are well-defined as a result of Ramsey s theorem Ram . Using Definition we make the following definition. THE ELECTRONIC JOURNAL OF COMBINATORICS 9 2002 R13 1 Definition A Ramsey r-coloring for R R k1 k2 . kr is an r-coloring of the complete graph on V R vertices which does not admit any monochromatic Kkj subgraph of color j for j 1 2 . r. For V R 1 we call the coloring a maximal Ramsey r-coloring. 2. THE LOWER BOUNDS We start with an easy bound which nonetheless improves upon some current best lower bounds. Theorem Let r 3. For any ki 3 i 1 2 . r we have R k1 k2 . kr k 1 1 R 2 k i . kr 1 . Proof. Let f G be a maximal Ramsey r 1 -coloring for R k2 k3 . kr with colors 2 3 . r. Let k1 3. Define graphs Gi i 1