tailieunhanh - Báo cáo toán học: "Fans and Bundles in the Graph of Pairwise Sums and Product"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Fans and Bundles in the Graph of Pairwise Sums and Products. | Fans and Bundles in the Graph of Pairwise Sums and Products Lorenz Halbeisen Department of Pure Mathematics Queen s University Belfast Belfast Northern Ireland halbeis@ Submitted Jan 15 2003 Accepted Oct 28 2003 Published Jan 2 2004 2000 Mathematics Subject Classification 11B75 05D10 11D99 Key-words Diophantine equations Pythagorean triples triangular numbers Abstract Let G be the graph on the vertex-set the positive integers N with n joined to m if n m and for some x y E N we have x y n and x y m . A pair of triangles sharing an edge . a K4 with an edge deleted and containing three consecutive numbers is called a 2-fan and three triangles on five numbers having one number in common and containing four consecutive numbers is called a 3-fan. It will be shown that G contains 3-fans infinitely many 2-fans and even arbitrarily large bundles of triangles sharing an edge. Finally it will be shown that GỊ 4. 1 Motivation If we colour the positive integers N with finitely many colours then by Ramsey s Theorem we find two one-to-one sequences xn n 1 and ynỴ i in N such that each set xn xm n m E N n III and yn ym n m E N n m is monochromatic. On the other hand it is known cf. HS Chapter that one can colour the positive integers with finitely many colours in such a way that there is no one-to-one sequence xn n 1 such that xn xm n m E N n m u xn xm n m E N n m is monochromatic. However it is not known if for any finite colouring of N there are distinct x and y in N such that x y and x y are monochromatic see also HS Question . Moreover it is not even known if there are x and y in N not both equal to 2 such that x y and x y are monochromatic. THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2004 R6 1 Let us state this problem in terms of graphs Let G N E be the graph on the vertex-set the positive integers N with n m E E if n m and for some x y E N we have n x y and m x y. Notice that for all n E N n n 1 E E this is just because n 1 n and n 1 1 n . If we colour