The generalized Erdős–Heilbronn (GEH) theorem, 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 by Balister and Wheeler in 2009.

Problem 1. Is the generalized Erdős–Heilbronn problem also true for any finite groups? In particular, is it true for finite cyclic groups?

This question is inspired by the construction of a counter-example to some variants of Ramsey theorem. In the construction we may not need the full strength of the GEH, so a related question is:

Problem 2. Is there any weaker results to the GEH, which have already been proved?

3 d -> p

The generalized Erdős–Heilbronn (GEH) theorem, 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 $|\hat{k}X| \geq \min \lbrace k|X|-k^2+1 , d p \rbrace$ 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 by Balister and Wheeler in 2009.

Problem 1. Is the generalized Erdős–Heilbronn problem also true for any finite groups? In particular, is it true for finite cyclic groups?

This question is inspired by the construction of a counter-example to some variants of Ramsey theorem. In the construction we may not need the full strength of the GEH, so a related question is:

Problem 2. Is there any weaker results to the GEH, which have already been proved?

2 typos, math bracket

# Is the generalized Erdős–Heilbronn problem true for finite cyclic groups?

The generalized Erdős–Heilbronn (GEH) theorem, 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 $|\hat{k}X| \geq \min{ min \lbrace k|X|-k^2+1 , d}$ d \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 $X$ is as a subset of a general any finite group $G$; the result is proved by Balister and Wheeler in 2009.

Problem 1. Is the generalized Erdős–Heilbronn problem also true for any finite groups? In particular, is it true for finite cyclic groups?

This question is inspired by the construction of a counter-example to some variants of Ramsey theorem. In the construction we may not need the full strength of the GEH, so a related question is:

Problem 2. Is there any weaker results to the GEH, which has have already been proved?

1