Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers of K3. | A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers of K3 Geoffrey Exoo Department of Mathematics and Computer Science Indiana State University Terre Haute IN 47809 ge@judy. Submitted September 13 1994 Accepted September 18 1994 Abstract For k 5 we establish new lower bounds on the Schur numbers S k and on the k-color Ramsey numbers of K3. For integers m and n let m n denote the set i I m i n . A set S of integers is called sum-free if i j 2 S implies i j 2 S where we allow i j. The Schur function S k is defined for all positive integers as the maximum n such that 1 n can be partitioned into k sum-free sets. The k-color Ramsey number of the complete graph Kn often denoted Rk n is defined to be the smallest integer t such that in any k-coloring of the edges of Kt there is a complete subgraph Kn all of whose edges have the same color. A sum-free partition of 1 s gives rise to a K3-free edge k-coloring of Ks. by identifying the vertex set of Ks. with 0 s and by coloring the edge uv according to the set membership of u v . Hence Rk 3 S k 2. It is known that S 1 1 S 2 4 S 3 13 and S 4 44. The first three values are easy to verify the last one is due to L. D. Baumert 1 . The best previously published bounds for S 5 are 157 S 5 321 the lower bound was proved in 4 and the upper bound in 6 . For Ramsey numbers we know R2 3 6 and R3 3 17 the current bounds on R4 3 are 51 and 65 5 . Below we list the five sets of a sum-free partition of 1 160 Since the partition is symmetric i and 161 i always belong to the same set only the integers from 1 to 80 are listed. 1 THE ELECTRONIC JOURNAL OF COMBINATORICS 1 1994 R8 2 Set 1 4 5 15 16 22 28 29 39 40 41 42 48 49 59 Set 2 2 3 8 14 19 20 24 25 36 46 47 51 62 73 Set 3 7 9 11 12 13 17 27 31 32 33 35 37 53 56 57 61 79 Set 4 1 6 10 18 21 23 26 30 34 38 43 45 50 54 65 74 Set 5 44 52 55 58 60 63 64 66 67 68 69 70 71 72 75 76 77 78 80 This proves that S 5 160. It follows that R5 3 162. From 1 and 2 we have S k c 321 k 5
