Erdős Problem 788 — strengthened final theorem

For each $n\in\mathbb N$, let $I_n=\{m\in\mathbb N:n<m<2n\}$ and $J_n=\{m\in\mathbb N:2n<m<4n\}$, and define $f(n)$ to be, viewed first as a natural number and then as an integer, the greatest $t\le |J_n|+|I_n|$ such that for every finite set $B\subseteq J_n$, there exists a finite set $A\subseteq I_n$ satisfying $x+y\notin B$ for every pair of distinct $x,y\in A$, and $t\le |B|+|A|$; here $B$ and $A$ may be empty, and the condition does not restrict sums $x+x$. The theorem asserts all of the following simultaneously: for every $n\in\mathbb N$ with $n\ge3$, $\frac1{2000}\sqrt{n\log n}\le f(n)$; there exist a real $K>0$ and a natural number $n_0\ge1$ such that, for every natural $n\ge n_0$, $\frac1{2000}\sqrt{n\log n}\le f(n)\le n^{\,1/2+K\left(\frac{\log(\log n)}{\log n}\right)^{1/3}}$; for every real $\varepsilon>0$, there exists a natural number $n_0\ge1$ such that every natural $n\ge n_0$ satisfies $n^{\,1/2-\varepsilon}\le f(n)\le n^{\,1/2+\varepsilon}$; and, as a separate conjunct with its own threshold witness, for every real $\varepsilon>0$, there exists a natural number $n_0\ge1$ such that every natural $n\ge n_0$ satisfies $f(n)\le n^{\,1/2+\varepsilon}$. All displayed occurrences of $f(n)$ are its integer value coerced to $\mathbb R$, the logarithms, square roots, division, and real powers are the corresponding total real operations, and the implications impose no condition on natural numbers below $3$ or below their respective existentially chosen thresholds.