Finite concatenation-free languages

Suppose, $$A$$ is a finite alphabet. $$L \subset A^*$$ is a language. Let's call $$L$$ concatenation-free iff $$\forall u, v \in L$$ we have $$uv \notin L$$.

Does there exist some function $$c: \mathbb{N} \to (0; 1)$$, such that for any finite language $$L \subset A^*$$, there exists a concatenation-free sublanguage $$L_0 \subset L$$, such that $$|L_0| \geq c(|A|)|L|$$?

The only thing I currently know about this problem, is that we can take $$c(1) = \frac{1}{3}$$. That is a direct consequence of Erdos-Sidon theorem, that states:

$$\forall A \subset \mathbb{Z}$$ $$\exists$$ a sum-free $$A_0 \subset A$$, such that $$|A_0| \geq \frac{|A|}{3}$$

However, I do not know how to deal with $$|A| \geq 2$$.