1. Schur Number of Groups Yusheng Li Ko-Wei Lih

2. Multi-Color Ramsey Numbers Define to be the minimum such that any edge coloring of in colors, there is a monochromatic

3. General Bounds

4. Schur Numbers for Integers A set S is called sum-free if for, where x,y are not necessarily distinct. Schur number is defined as the smallest N such that can be partitioned into k sum-free sets. For any, the equality holds for k=1,2.

5. Some Schur Numbers No many Schur numbers are known. For k=3, there is a gap between and, and the gap even larger for k=4.

6. Schur Numbers for Groups Let H be a group, and let Let be the largest cardinality of sum-free set in and let be the smallest k such that can be partitioned into k sum-free sets. Let

7. Generalized Schur Numbers Generally,, and Moreover, the equality holds for k=1,2,3.

8. Partition Some Groups I Partition into three sum-free sets This gives as desired. (0,1) (0,2) (1,2) (2,0) (1,0) (2,1) (0,3) (1,1) (2,2) (1,3) (3,0) (2,3) (3,1) (3,3) (3,2)

9. Partition Some Groups II Let be Klein four-element group. Then can be partitioned three sum- free sets. (0,a) (0, b) (0,c) (a,0) (b, 0) (c, 0) (a, b) (b, c) (c, a) (b, a) (c, b) (a, c) (b, b) (c, c) (a, a)

10. Partition Some Groups III Klein group is a product, so can be partitioned into three sum-free sets. (0,0,0,1)(0,0,1,0)(0,0,1,1) (0,1,0,0)(1,0,0,0)(1,1,0,0) (0,1,1,0)(1,0,1,1)(1,1,0,1) (1,0,0,1)(1,1,1,0)(0,1,1,1) (1,0,1,0)(1,1,1,1)(0,1,0,1)

11. Recursive Upper Bound It is easy to see from a partition where are sum-free sets of.

12. Partition Product of Binary Groups Problem: Find constant c, as small as possible, such that The current c is ¾ basing on the fact that. Is it possible that

13. Definition for Finite Fields Let F be a finite field. Define to be the smallest index of multiplicative subgroup A of such that A is sum-free.

14. Computing for Small Fields I p2357111317192329 12235349117 p31374143475359616771 109462315296225 p7379838997 101103107109113 9264111653453127

15. Computing for Small Fields More values of as follows. m\p235711131719 23886158329 372662183536 12282286 4316 246042192144 m\p23293137414347 2332115366042138 377364 3310 108 1723 126 1702 4264210480720840 21001472

16. End We are in hardness to find Thank you