Inspired by this question and remembering Bertrand's postulate, I wondered.
Does any set of random odd numbers satisfying Bertrand's postulate satisfy Goldbach's conjecture (i.e. that every even number can be written as the sum of two numbers in the set)?
In short, the answer is NO.
However, looking up Bertrand's postulate I found that there have since been many improvements on the bounds.
As a set I took all primes up to $p=2010881$$=\textrm{nextprime}(2010760)$ and afterwards generated odd numbers $o$ satisfying $n < o < (1+\frac{1}{16597})n$ up to $2^{30}$.
Surprisingly, this set satisfied Goldbach's conjecture (for $n < 2^{30}$) while the set is a lot smaller than the set of actual primes.
Since then I have generated $4$ more of these sets, each one satisfying Goldbach's conjecture up to at least $n < 2^{30}$.
My question is if there are counterexamples to this idea (that any such set satisfies Goldbach's conjecture)? If there are, what about some even stronger results of the postulate? If not, a proof of this would also prove Goldbach's conjecture.
Searching for $n < 2^{32}$ I have found a counterexample to the above question.
I have also found counterexamples performing this search with the $n < o \leq \left( 1 + \frac{1}{\ln^3(x)} \right)n$ and the $n < o \leq \left( 1 + \frac{1}{25\ln^2(x)} \right)n$ criterion.
However, $n < o \leq \left( 1 + \frac{1}{5000\ln^2(x)} \right)n$ remains without finding any counter-example.