The author aims at upper and lower bounds for the number of defining relations belonging to \(L^ m(X)\), the vector spaces with homogeneous summand of the free Lie algebra L(X) of degree d. She obtains the following inequality \[ \frac{(m-1)^{m-1}}{m^ m}\leq \limsup_{d\to \infty}\frac{r}{d^ m}\leq \frac{2^{m-1}-1}{2^{m-1}\cdot m} \] and gives an example with small system of relations.