GPT 6 Astra Ultra attempt on Math-726

Erdős Problem #241: independent continuation

Date: 2026-09-23. Research attempt; no claim of a new published result.

1) TARGET AND SOURCE AUDIT

If $f(N)$ is the largest cardinality of a $B_3$-set in $[1,N]$, is $f(N)\sim N^{1/3}$?

All supplied versions considered: 241.tex. The source supplies the known asymptotically sharp lower bound and quotes a stronger literature upper constant. I give an independent elementary upper-bound audit using mixed sums; this does not improve that quoted literature constant.

2) WORK

Let $A\subset[1,N]$ be $B_3$ and $m=|A|$. Consider all expressions $a+b-c$ with $a\le b$ in $A$ and $c\in A\setminus\{a,b\}$. They are pairwise distinct. Indeed an equality gives \[ a+b+f=d+e+c. \] Triple-sum uniqueness gives equality of the multisets $\{a,b,f\}$ and $\{d,e,c\}$. Since $c$ is neither $a$ nor $b$, it must equal $f$; cancellation then gives $\{a,b\}=\{d,e\}$. There are $m\binom m2=m^2(m-1)/2$ such expressions: fix $c$, then choose a two-element multiset from the remaining $m-1$ elements. They all lie in $[2-N,2N-1]$.

Moreover none belongs to $A$. If $a+b-c=t\in A$, then $a+b=c+t$, and $B_3$ implies $B_2$ by adding any fixed member of $A$ to both sides. Thus $\{a,b\}=\{c,t\}$, impossible. Consequently \[ \frac{m^2(m-1)}2+m\le3N-2, \] which implies $m\le(6N)^{1/3}+O(1)$. This improves the most naive triple-counting constant $18^{1/3}$, but is weaker than the literature constant $(7/2)^{1/3}$ already quoted in the source.

3) ADVERSARIAL VERIFICATION

Repeated $a=b$ is allowed and included in the count; $c$ is deliberately excluded from that multiset. Without the exclusion, the map has many collisions. All expressions need not be positive, so the ambient interval must have length about $3N$, not $N$.

4) FINAL

UNRESOLVED. The gap from the available upper constant to one is untouched. No sharp asymptotic or novelty claim follows from this elementary mixed-sum argument.

5) SOURCES CHECKED

No external theorem is used in the new argument. The supplied source was read in full; the saved Tao status snapshot is dated 2026-09-22.

6) COMPLETION ESTIMATE

This is a conservative subjective measure of progress toward the original question, not a probability of correctness or a claim that the remaining work is routine. COMPLETION: 12%