Coprime graph of a dense subset of $[n]$: does the extremal threshold force all short odd cycles? (Erdős #883)

For $A\subseteq\{1,\ldots,n\}$ let $G(A)$ be the coprime graph: its vertices are the elements of $A$, and two integers are joined by an edge exactly when they are coprime. Write $T(n)=\lfloor\tfrac{n}{2}\rfloor+\lfloor\tfrac{n}{3}\rfloor-\lfloor\tfrac{n}{6}\rfloor$ (the number of integers in $[1,n]$ divisible by $2$ or $3$). Question (1): is it true that if $\lvert A\rvert>T(n)$ then $G(A)$ contains every odd cycle of length $\leq \tfrac{n}{3}+1$? Question (2): is it true that for every fixed $\ell\geq 1$, whenever $n$ is sufficiently large and $\lvert A\rvert>T(n)$, the graph $G(A)$ must contain a complete tripartite graph $K_{1,\ell,\ell}$ (on $2\ell+1$ vertices)?

Acceptance. FULLY RESOLVES: settle question (1), the sharp odd-cycle claim — either a complete proof that for all sufficiently large $n$, every $A\subseteq[n]$ with $\lvert A\rvert>T(n)$ has $G(A)$ containing all odd cycles of length $\leq \tfrac{n}{3}+1$; or a finite, machine-checkable counterexample: an explicit $n$ and set $A\subseteq[n]$ with $\lvert A\rvert>T(n)$ whose coprime graph $G(A)$ omits some odd cycle of length $\leq \tfrac{n}{3}+1$ (deliver $A$, verify $\lvert A\rvert>T(n)$, and certify the missing cycle length by exhaustive cycle search). A proof should be machine-checkable where feasible. ADVANCES (each independently checkable): (a) improve the odd-cycle constant $c$ in the Erdős–Sárközy theorem strictly beyond the value implicit in the background, toward the conjectured $c=1/3$, with a complete proof; (b) exhaustively verify question (1) for a new range of $n$ with reproducible code and an exhaustiveness certificate; or (c) exhibit near-extremal sets $A$ that pin down bounds on the true constant. Deliver the proof, the counterexample set + cycle certificate, or the search code + verified range.