Unverified LLM-generated attempt. Automated screening checks form, honesty and scope, not correctness. Status at erdosproblems.com: open. Claims here do not change the source status.
Erdős #201; independent attempt, 2026-09-23. For $|B|=N$, let $f_3(B)$ be its maximum three-term-progression-free subset size; set $G_3(N)=\min f_3(B)$ and $R_3(N)=f_3([N])$. The target is the sharp comparison $R_3(N)/G_3(N)\to1$. I read all of 201.tex. It invokes O'Bryant's progression-type theorem. Here I obtain a self-contained necessary condition on any putative extremal set, without importing that theorem.
WORK: COUNTING THE REQUIRED OBSTRUCTIONS
Let $T$ count nontrivial three-term progressions of $B$ as unordered triples, and write $r=f_3(B)$. Independently retain each element with probability $p$. Deleting at most one point from each surviving progression leaves a progression-free set, so \[
r\ge pN-p^3T\qquad(0\le p\le1).
\] If $T<N/3$, taking $p=1$ gives $r>N(2/3)$. Therefore whenever $r\le2N/3$ we have $T\ge N/3$ and may put $p=\sqrt{N/(3T)}\le1$. This yields \[
r\ge\frac{2}{3\sqrt3}\frac{N^{3/2}}{\sqrt T},
\qquad
T\ge\frac{4N^3}{27r^2}.
\] More generally, for $k$-term progressions, optimizing $pN-p^kT_k$ gives \[
f_k(B)\ge\frac{k-1}{k}N\left(\frac{N}{kT_k}\right)^{1/(k-1)}
\quad\text{when }T_k\ge N/k.
\] Thus a proposed family with unusually small progression-free subsets must support quantitatively many actual progressions. This is the usual alteration argument, not a novel extremal theorem.
ADVERSARIAL VERIFICATION AND REMAINING GAP
A progression is counted once, by its increasing ordering; arbitrary ordered endpoint counts would change constants. Deleting a vertex cannot create new progressions, so overlapping obstructions cause no problem. The cutoff $T\ge N/3$ is essential to keep $p\le1$. A large $T$ is only necessary: it gives no matching upper bound on $f_3(B)$ and no characterization of sets minimizing it. The supplied claim that low-type families cannot be extremal requires comparison to an appropriate upper bound for $R_3$, not merely to another lower bound. No ratio-one conclusion follows from either displayed alteration inequality.
SOURCES CHECKED
Complete local source: gpt_pro_5.4/201.tex. The source's stated ratio problem is used as the target. An external indexed-statement search was attempted on 2026-09-23; no additional external theorem is relied upon in this self-contained calculation.
FINAL
UNRESOLVED. A necessary quantitative progression-count condition is verified, but the sharp comparison remains open in this attempt.
COMPLETION ESTIMATE
Subjective progress toward the ratio-one target. COMPLETION: 8%