The [generalized Erdős–Heilbronn (GEH) theorem](http://en.wikipedia.org/wiki/Restricted_sumset), which is proved by da Silva and Hamidoune in 1994, states that:


**Theorem.** If p is a prime and $X$ is a subset of $\mathbb{Z}\_p$, then $|\hat{k}X| \geq \min \lbrace k|X|-k^2+1 , p \rbrace$ for $\hat{k}X = \lbrace x\_1+\ldots+x\_k \mid x\_i \in X , x\_i \neq x\_j \rbrace$.


Also, for the case that $k=2$ (which is called the Erdős–Heilbronn problem), the above statement holds for $X$ as a subset of any finite group $G$; the result is [proved](http://jeffreypaulwheeler.com/Final%20Erdos-Heilbronn_Problem.pdf) by Balister and Wheeler in 2009.



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> **Problem 1.** Is the generalized Erdős–Heilbronn problem also true for any finite groups? In particular, is it true for finite cyclic groups?
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This question is inspired by the construction of a [counter-example to some variants of Ramsey theorem](https://cstheory.stackexchange.com/questions/3795/extensions-of-ramseys-theorem-monochromatic-but-diverse/3803#3803). In the construction we may not need the full strength of the GEH, so a related question is:



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> **Problem 2.** Is there any weaker results to the GEH, which have already been proved?
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Is the generalized Erdős–Heilbronn problem true for finite cyclic groups?
Arithmetic progressions usually give small sumsets so...How about $X=\lbrace 0,3,6,9,12 \rbrace \subset \mathbb{Z}\_{15}$? Then for $k=2,3,4$ one has $\hat{k}X=X$. That is ok for $k=4$ but not for $k=2,3$. (Doesn't that contradict what you said about $k=2$?)


In fact $X=\lbrace 0,d,2d,\cdots,nd\rbrace \subset \mathbb{Z}\_{nd+d}$ for $n \ge 5$ is always a counter-example for $k=3$.


The same type of examples should work as well for larger $k$ and should also work with a few elements removed.


**later** One can't hope for better than $|\hat{k}X| \geq \min \lbrace k|X|-k^2+1 , |H| \rbrace$ where $H$ is the subgroup generated by $X$. After all, $\hat{k}X \subset H$. I don't know if that is true though. A weaker claim, which seems somewhat more likely, is to replace $|H|$ by the size of the smallest subgroup of size at least $|X|$. I have not read the Balister and Wheeler paper you mention but it seems (from the review posted) to replace $|H|$ with the size of the smallest non-trivial subgroup, which would be prime of cyclic order.


