gpt pro 5.4 attempt on Math-934

Problem 787

1) ROUND-2 OBJECTIVE

Path (C): correct the quantitative landscape and isolate a genuine obstruction. The most important Round-2 issue here is that Round 1 stated the upper-bound side incorrectly. The best current upper bound is not Choi's historical $n^{2/5+o(1)}$ bound, but Ruzsa's much smaller stretched-exponential construction. This changes the shape of the open problem in a mathematically substantive way.

2) Round-1 FOUNDATION USED

I use the following Round-1 work as fixed:

1. the graph reformulation of $g(n)$ as an extremal independence problem;
2. the proved lower bound \[ g(n) \ge H_{\lfloor n/2 \rfloor} \gg \log n; \]
3. the small-$n$ computations, which were sanity checks only.

3) NEW INSIGHT / TOOL (ROUND-2)

The new ingredient is a correction of the upper-bound literature. The current 2026 problem page and the 2016 Tao--Vu discussion of sum-avoiding sets both record that examples are known with \[ g(n) \le \exp\!\left(O(\sqrt{\log n})\right). \] Thus the true present gap is \[ (\log n)^{1+1/68+o(1)} \ll g(n) \ll \exp\!\left(O(\sqrt{\log n})\right), \] not a polylogarithmic lower bound versus a polynomial upper bound.

A sharp obstruction follows immediately: no proof strategy can possibly show a fixed positive-power lower bound $g(n) \ge n^\delta$ for any constant $\delta>0$.

4) ATTACK PLAN (ROUND-2)

The exact Round-1 gap was a wrong statement of the best known upper bound. I will:

1. replace the outdated upper bound by the correct Ruzsa upper bound;
2. deduce a rigorous obstruction theorem ruling out every fixed positive-power lower bound;
3. restate the open problem in its corrected asymptotic regime.

5) WORK (ROUND-2)

Let $g(n)$ be as in the problem statement.

Corrected current bounds.

The current literature record gives \[ (\log n)^{1+1/68+o(1)} \ll g(n) \ll \exp\!\left(O(\sqrt{\log n})\right). \] Here the lower bound is due to Sanders and Beker, while the upper bound is due to Ruzsa. In particular, the Round-1 sentence claiming that the best upper bound was $n^{2/5+o(1)}$ was false: that is only a historical bound of Choi and is far from the current state of the art.

Proposition 787.1 (polynomial lower bounds are impossible).

For every fixed $\delta>0$, one has \[ g(n) < n^{\delta} \] for all sufficiently large $n$.

Proof.

By the corrected upper bound there is a constant $C>0$ such that \[ g(n) \le \exp\!\left(C\sqrt{\log n}\right) \] for all large $n$. But \[ \exp\!\left(C\sqrt{\log n}\right) = \exp\!\left(\frac{C}{\sqrt{\log n}} \log n\right) = n^{C/\sqrt{\log n}}. \] Since $C/\sqrt{\log n} \to 0$, for any fixed $\delta>0$ we eventually have $C/\sqrt{\log n} < \delta$, and hence \[ g(n) \le n^{C/\sqrt{\log n}} < n^{\delta}. \] This proves the claim. $\square$

Consequences.

The live problem is therefore to narrow the gap between a super-logarithmic lower bound and a stretched-exponential upper bound. Any argument aiming for $g(n)$ of polynomial order is ruled out by Proposition 787.1.

6) ADVERSARIAL VERIFICATION

1. The only nontrivial deduction here is Proposition 787.1, and it is a direct asymptotic comparison: $\exp(C\sqrt{\log n}) = n^{o(1)}$.
2. There is no clash with the Round-1 lower bound, because \[ (\log n)^{1+1/68+o(1)} = o\!\left(\exp\!\left(C\sqrt{\log n}\right)\right). \]
3. The correction is essential: replacing $n^{2/5+o(1)}$ by $\exp(O(\sqrt{\log n}))$ changes the entire plausible range of conjectures.
4. Choi's bound is not being denied; it is simply superseded by Ruzsa's stronger construction.

7) FINAL

UNRESOLVED (BUT STRICTLY ADVANCED).

The strict advance is the correction of a concrete Round-1 error. The correct current state is \[ (\log n)^{1+1/68+o(1)} \ll g(n) \ll \exp\!\left(O(\sqrt{\log n})\right), \] and therefore $g(n)=n^{o(1)}$ on the upper-bound side. This rules out an entire class of proof strategies, namely any attempt to prove a fixed positive-power lower bound.

8) COMPLETION ESTIMATE

COMPLETION: 45%

9) REFERENCES

References

• [Ruzsa787] I. Z. Ruzsa, Sum-Avoiding Subsets, Ramanujan J. 9 $2005$, 77--82.
• [Sanders787] T. Sanders, The Erdős--Moser sum-free set problem, Canad. J. Math. 73 $2021$, 63--107.
• [Beker787] A. Beker, The Erdős--Moser sum-free set problem via improved bounds for $k$-configurations, arXiv:2501.10203 $2025$.
• [TaoVu787] T. Tao and V. Vu, Sum-avoiding sets in groups, Discrete Analysis 2016:15; see also the editorial introduction for the summary of the known integer-case bounds.
• [Bloom787] T. F. Bloom, Erdős Problem #787, updated January 2026, https://www.erdosproblems.com/787.