Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: An Improvement to Mathon’s Cyclotomic Ramsey Colorings. | An Improvement to Mathon s Cyclotomic Ramsey Colorings Xiaodong Xu Guangxi Academy of Sciences Nanning 530003 China xxdmaths@ Stanislaw P. Radziszowskiy Department of Computer Science Rochester Institute of Technology Rochester NY 14623 USA spr@ Submitted Oct 28 2008 Accepted Dec 9 2008 Published Jan 7 2009 Mathematics Subject Classification 05C55 Abstract In this note we show how to extend Mathon s cyclotomic colorings of the edges of some complete graphs without increasing the maximum order of monochromatic complete subgraphs. This improves the well known lower bound construction for multicolor Ramsey numbers in particular we obtain R3 7 3214. 1 Introduction and Notation A k1 k2 . k m -coloring for integers m kị 1 is an assignment of one of m colors to each edge in a complete graph such that it does not contain any monochromatic complete subgraph Kki in color i for 1 i m. Similarly a ki k2 . km n -coloring is a k1 . km -coloring of the complete graph on n vertices Kn. Let R k1 . km and R k1 . km n denote the set of all k1 . km - and k1 . km n -colorings respectively. The Ramsey number R k1 . km is defined to be the least n 0 such that partially supported by the National Science Fund of China 60563008 and the Basic Research Fund of Guangxi Academy of Sciences 080414 y corresponding author THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 N1 1 R k1 . km n is empty. In the diagonal case where k1 . km k we will use simpler notation Rm k and Rm k n for sets of colorings and Rm k for the Ramsey numbers. In the case of 2 colors m 2 we deal with classical graph Ramsey numbers which have been studied extensively for 50 years. Much less has been done for multicolor numbers m 3 . A related area of interest has been the study of generalized Ramsey colorings wherein the forbidden monochromatic subgraphs are not restricted to complete graphs. The second author maintains a regularly updated survey 2 of the most recent results on the best known bounds on various
