Upper bound for size of subsets of a finite group that contains a sum-full set

Question

Problem

I'm looking for an upper bound for the number $k(G)$ of a finite group $G$, defined as follow:

Let $\mathcal{F}_k$ be the family of subsets of $G$ with size $k$, and we define $k(G)$ be the minimum $k$ such that every subset $X \in \mathcal F_k$ contains a non-empty sum-full set $S$, which is a set satisfies $$ S \subseteq S+S := \{ x+y \mid x,y \in S \}. $$

Note that the inequality $k(G) \leq |G|$ holds trivially since there is only one subset in $\mathcal F_{|G|}$ which is $G$ itself, and $G$ is a semigroup indeed.

Are there any papers or references about this number $k(G)$? Does it have a name? I'm interesting in particularly upper bounds of $k(G)$, but any related results are fine.

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Motivation

The restricted Davenport number $\hat{D}(G)$ of a group $G$, is defined as the smallest number $d$ such that given a subset $A \in \mathcal F_d$, there exists a zero-sum non-empty subset $S \subseteq A$, that is,

$$ \sum_{x \in S} x = 0, $$

where $0$ is the identity in $G$. In the paper "On a conjecture of Erdos and Heilbronn", Szemeredi has proved:

$$\hat{D}(G) = O(\sqrt{|G|}). $$

Hamidoune and Zemor set a precise bound $\sqrt{2}$ on the constant of the big-O notation.

I'm trying to provide a link between $\hat{D}(G)$ and the number $k(G)$; it seems to me that the size of sum-full sets in $G$ may related to the zero-sum problem. I'll provide the justification in another post, which is highly related.

Answer 1

First of all the $k(G)$ cannot be smaller than the size of any proper subgroup of $G$, because if $H$ is a proper subgroup, $gH$ is a coset, $g\not\in H$, then $gHgH$ does not intersect $gH$ (if $ghgh'=gh''$, then $g\in H$). For example, if $G$ is Abelian, $|G|$ is not prime (i.e. $G$ is not cyclic of prime order), then $k(G)>\sqrt{|G|}$: look at the size of a maximal proper subgroup.