tailieunhanh - Báo cáo toán học: "Generalizing the Ramsey Problem through Diameter"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Generalizing the Ramsey Problem through Diameter | Generalizing the Ramsey Problem through Diameter Dhruv Mubayi Submitted January 8 2001 Accepted November 13 2001. MR Subject Classifications 05C12 05C15 05C35 05C55 Abstract Given a graph G and positive integers let fd G be the maximum t such that every k-coloring of E G yields a monochromatic subgraph with diameter at most d on at least t vertices. Determining ff Kn is equivalent to determining classical Ramsey numbers for multicolorings. Our results include determining fk Ka b within 1 for all for d 4 f3 Kn n 2 1 or n 2 depending on whether n 2 mod 4 or not fk Kn The third result is almost sharp since a construction due to Calkin implies that fk Kn k i k 1 when k 1 is a prime power. The asymptotics for fk Kn remain open when d k 3 and when d 3. k 4 are fixed. 1 Introduction The Ramsey problem for multicolorings asks for the minimum n such that every k-coloring of the edges of Kn yields a monochromatic Kp. This problem has been generalized in many ways see . 2 6 7 9 12 13 14 . We begin with the following generalization due to Paul Erdos 8 see also 11 Problem 1 What is the maximum t with the property that every k-coloring of E Kn yields a monochromatic subgraph of diameter at most two on at least t vertices A related problem is investigated in 14 where the existence of the Ramsey number is proven when the host graph is not necessarily a clique. Call a subgraph of diameter at most d a d-subgraph. Department of Mathematics Statistics and Computer Science University of Illinois at Chicago 851 S. Morgan Street Chicago IL 60607-7045 mubayi@ Keywords diameter generalized ramsey theory THE ELECTRONIC JOURNAL OF COMBINATORICS 9 2002 R41 1 Theorem 2 Tonoyan 14 Let D k 1 d D n 2. Then there is a smallest integer t RD k n d such that every graph G with diameter D on at least t vertices has the following property every k-coloring of E G yields a monochromatic d-subgraph on at least n vertices. We study a problem closely related to Tonoyan s result .