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Claim. Ritvik Nayak's note Incident-GCD Bounds for an Erdős-Szemerédi LCM Block Problem (dated April 2026), posted on 29 April 2026 in the discussion thread of Problem 873, proves in Theorems 6.1 and 7.2 that for every increasing sequence $A$

$$F(A,X,11)\ll X^{223/676}\log X,\qquad F(A,X,6)\ll X^{437/1320}\log X .$$

Both exponents lie below the exponent $1/3$ of the Erdős–Szemerédi bound for $k=3$. The argument combines a Vandermonde lower bound for least common multiples with the gcd structure forced inside short windows near the scale $a_i\asymp X^{1/2}$, which gives lower bounds for the span of such windows and, by greedy selection, the counting bounds. The note presents itself as partial progress, not a solution. The post says that GPT 5.4 Thinking helped with algebraic notation and with checking the argument; the note describes partial GPT-based assistance for checking algebra, editing and formatting.

Submission note. Posted to the site's forum by Ritvik Nayak on 29 April 2026:

I would like to share my partial result on this problem.

This partial result proves two unconditional bounds that go beyond the usual triple-block exponent. In particular, it proves

$$F(A,X,11)\ll X^{223/676}\log > X$$

and

$$F(A,X,6)\ll X^{437/1320}\log X.$$

Both exponents are strictly

smaller than $1/3$, so the result shows that the classical triple-block barrier can be crossed for longer consecutive blocks.

The main idea is to combine the Vandermonde lcm inequality with gcd structure forced inside compact near-critical blocks. Around the difficult scale $a_i\asymp X^{1/2}$, a good block must contain substantial pairwise gcd structure. The argument organizes this information through incident-star and complete-edge configurations, which then give stronger span bounds and lead to the counting estimates above.

I view this as genuine partial progress on the problem, rather than a solution of the full $X^\varepsilon$ question. I used GPT 5.4 Thinking for helping me with algebraic notation and verifying the contribution, and it passed. I would be very grateful for any corrections, comments, or references to related work that I may have missed.

Covers. The question for every $\varepsilon>223/676$, with $k=11$ and $X$ large, and for every $\varepsilon>437/1320$ with $k=6$. Not covered: $\varepsilon\le223/676$. The later claim of old-bielefelder covers every $\varepsilon>1/4$.

Standing. Claimed: the note has no arXiv or journal record. A reply of 29 April 2026 reports smaller exponents for $k=6$ and $k=8$ from a GPT-5.5 Pro chat without a manuscript, and gets no page; another reply reports an automated check, which is a thread comment, not a review. The site's label is OPEN.

Depends on. No page of this wiki.