Dense infinite $B_h$ sequences

For $h=3$ and $h=4$ we prove the existence of infinite $B_h$ sequences $\B$ with counting function $\mathcal{B}(x)= x^{\sqrt{(h-1)^2+1}-(h-1) + o(1)}.$ This result extends a construction of I. Ruzsa for $B_2$ sequences.

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