Schur Numbers for Sum-Free Sets in Groups

Yusheng Li 1 and Ko-Wei Lih 2
1 Department of Mathematics, Tongji University, Shanghai 200092, P. R. China
2 Institute of Mathematics, Academia Sinica, Nankang, Taipei 115, Taiwan

Abstract     Full Text  PPT

Let $N\ge 0$ be an integer and let $[N]=\{0,\,1,\ldots,\,N\}$. Let $H$ be a product of an additive group and an $[N]$ and let $H^{*}=H\setminus\{0\}$. Define $\chi(H)$ to be the smallest $k$ such that $H^{*}$ can be partitioned into $k$ sum-free sets, and define $t_k$ be the maximum $|H|$ such that $\chi(H)=k$. It is shown that $r_k(3)\ge t_k+1$, where $r_k(3)$ is the $k$-color Ramsey number of triangle and the equality holds for $k=1,\,2,\,3$. We also compute some values of $\chi_0(F)$, where $F$ is a finite field and $\chi_0(F)$ is the smallest index of sum-free multiplicative subgroup of $F^{*}$, and give some upper bounds of $t_k$ by improving the known upper bounds of $r_k(3)$.