Schur Number of Groups

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.

9. Partition Some Groups II • Let be Klein four-element group. Then can be partitioned three sum-free sets.

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

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

15. Computing for Small Fields • More values of as follows.

16. End • We are in hardness to find • Thank you