Is Fibonacci sequence the minimum of unique pairwise sum sequence? Let $(a_n)_{n=1}^\infty$ be a strictly increasing (condition added per earlier answer of Amitesh Datta) sequence of natural numbers where all pairwise element sums are unique. Can anyone prove or disprove whether the Fibonacci sequence $(f_n)_{n=1}^\infty=(1,2,3,5,8,\cdots)$ is the "minimum" of such sequences, i.e., $f_n\le a_n$, for all such sequences $(a_n)$?

No.  The Mian-Chowla sequence is such a sequence and it begins 1, 2, 4, 8, 13... ; it also grows only polynomially fast, no faster than n3.
