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For $h \geq 1$, a $B_h$-set is a set of integers such that every integer $n$has at most one representation in the form $n = a_{i_1} + \cdots + a_{i_h}$,where $a_{i_j} \in A$ for all $j = 1,\ldots, h$ and $a_{i_1} \leq \ldots \leqa_{i_h}$. The greedy $B_h$-set is the infinite set of nonnegative integers$\{a_0(h), a_1(h), a_2(h), \ldots \}$ constructed as follows: If $a_0(h) = 0$and $\{a_0(h), a_1(h), a_2(h), \ldots, a_k(h) \}$ is a $B_h$-set, then$a_{k+1}(h)$ is the least positive integer such that $\{a_0(h), a_1(h), a_2(h),\ldots, a_k(h), a_{k+1}(h) \}$ is a $B_h$ set. One has $a_1(h) = 1$ and $a_2(h)= h+1$ for all $h$. Elementary proofs are given that $a_3(h) = h^2+h+1$ for all$h \geq 1$ and that $a_k(h) \leq \sum_{i=0}^{k-1} h^i$ for all $h \geq 1$ and$k \geq 1$.

The third positive element in the greedy $B_h$-set