## Monochromatic and zero-sum sets of nondecreasing modified diameter.(English)Zbl 1084.05073

Summary: Let $$m$$ be a positive integer whose smallest prime divisor is denoted by $$p$$, and let $${\mathbb Z}_m$$ denote the cyclic group of residues modulo $$m$$. For a set $$B=\{x_1,x_2,\dots,x_m\}$$ of $$m$$ integers satisfying $$x_1<x_2<\cdots<x_m$$, and an integer $$j$$ satisfying $$2\leq j \leq m$$, define $$g_j(B)=x_j-x_1$$. Furthermore, define $$f_j(m,2)$$ (define $$f_j(m,{\mathbb Z}_m)$$) to be the least integer $$N$$ such that for every coloring $$\Delta : \{1,\ldots,N\}\rightarrow \{0,1\}$$ (every coloring $$\Delta : \{1,\dots,N\}\rightarrow {\mathbb Z}_m$$), there exist two $$m$$-sets $$B_1,B_2\subset \{1,\dots,N\}$$ satisfying: (i) $$\max(B_1)< \min(B_2)$$, (ii) $$g_j(B_1)\leq g_j(B_2)$$, and (iii) $$|\Delta (B_i)|=1$$ for $$i=1,2$$ (and (iii) $$\sum_{x\in B_i}\Delta (x)=0$$ for $$i=1,2$$). We prove that $$f_j(m,2)\leq 5m-3$$ for all $$j$$, with equality holding for $$j=m$$, and that $$f_j(m,{\mathbb Z}_m)\leq 8m+{m\over p}-6$$. Moreover, we show that $$f_j(m,2)\geq 4m-2+(j-1)k$$, where $$k=\left\lfloor\left(-1+\sqrt{{8m-9+j\over j-1}}\right){/2}\right\rfloor$$, and, if $$m$$ is prime or $$j\geq{m\over p}+p-1$$, that $$f_j(m,{\mathbb Z}_m)\leq 6m-4$$. We conclude by showing $$f_{m-1}(m,2)=f_{m-1}(m,{\mathbb Z}_m)$$ for $$m\geq 9$$.

### MSC:

 05D05 Extremal set theory 11B75 Other combinatorial number theory

### Keywords:

cyclic group; coloring
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