Schur Number of Groups Yusheng Li Ko-Wei Lih

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

General Bounds

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.

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.

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

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

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)

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)

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)

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

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

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.

Computing for Small Fields I p p p

Computing for Small Fields More values of as follows. m\p m\p

End We are in hardness to find Thank you