Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: New lower bound for multicolor Ramsey numbers for even cycles. | New lower bound for multicolor Ramsey numbers for even cycles Tomasz Dzido Institute of Mathematics University of Gdansk Wita Stwosza 57 80-952 Gdansk Poland. tdz@ Andrzej Nowiki Institute of Mathematics University of Gdannsk Wita Stwosza 57 80-952 Gdannsk Poland. nowik@ Piotr Szuca Institute of Mathematics University of Gdannsk Wita Stwosza 57 80-952 Gdansk Poland. pszuca@ Submitted Oct 5 2004 Accepted Jun 3 2005 Published Aug 30 2005 Mathematics Subject Classifications Primary 05C55 Secondary 05C15 05C38 Abstract For given finite family of graphs G1 G2 . Gk k 2 the multicolor Ramsey number R G1 G2 . Gk is the smallest integer n such that if we arbitrarily color the edges of the complete graph on n vertices with k colors then there is always a monochromatic copy of Gi colored with i for some 1 i k. We give a lower bound for k color Ramsey number R Cm Cm . Cm where m 4 is even and Cm is the cycle on m vertices. 1 Introduction In this paper all graphs considered are undirected finite and contain neither loops nor multiple edges. By Km we denote the complete graph on m vertices and by Cm we denote the cycle of length m. For given graphs G1 G2 . Gk k 2 the multicolor The first author was partially supported by KBN grant 4 T11C 047 25. Iphe second author was partially supported by grant BW 5100-5-0145-4. Iphe third author was partially supported by grant BW 5100-5-0095-5. THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 N13 1 Ramsey number R G1 G2 . Gk is the smallest integer n such that if we arbitrarily color the edges of the complete graph of order n with k colors then it always contains a monochromatic copy of Gi colored with i for some 1 i k. We denote such a number by Rk G if G G1 G2 Gk. Here in we consider only 3-color Ramsey number R3 G . we color the edges of the complete graph Kn with color red blue and green. A 3-coloring of Kn is called a G n 3-coloring if it contains neither a red G nor a blue G nor a green G
