群中的Sumfree集：一个综述
Sumfree sets in groups: a survey
摘要：
我们讨论了几个问题有关的总和自由集的群体，提出了Erdos在他的调查“极值问题的数论“（会议论文集。Pure Math. VIII AMS）出版于1965年。除其他事项外，我们给出了一个大集合$A$在阿贝尔群$G$不包含一个子集$B$的固定大小$k$，使$B$的任何两个不同的元素之和不属于$A$（换句话说，$B$是无和关于$A$）。在上述调查中，Erders指出，如果$|A|$与$k$相比足够大，则$A$包含两个相加为零的元素。已知这对于$k \leq 3$是正确的。我们给出了所有$k \ge 4$的反例。另一方面，利用新的刻画结果，我们能够证明一个积极的结果时，$|G|$是不能整除的小素数。
英文摘要：
We discuss several questions concerning sum-free sets in groups, raised by Erdős in his survey "Extremal problems in number theory" (Proceedings of the Symp. Pure Math. VIII AMS) published in 1965. Among other things, we give a characterization for large sets $A$ in an abelian group $G$ which do not contain a subset $B$ of fixed size $k$ such that the sum of any two different elements of $B$ do not belong to $A$ (in other words, $B$ is sum-free with respect to $A$). Erdős, in the above mentioned survey, conjectured that if $|A|$ is sufficiently large compared to $k$, then $A$ contains two elements that add up to zero. This is known to be true for $k \leq 3$. We give counterexamples for all $k \ge 4$. On the other hand, using the new characterization result, we are able to prove a positive result in the case when $|G|$ is not divisible by small primes.
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