On an extremal problem in number theory

We define h(n) to be the largest function of n such that from any set of n nonzero integers, one can always extract a subset of h(n) integers with the property that any two sums formed from its elements are equal only if they have equal number of summands. A result of Erdös implies that $\text{h(n) ? n}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}$, and it is the aim of the present paper to obtain the refinement $\text{h(n) ? n}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}\text{(}\text{log}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}$.