gpt pro 5.2 attempt on Math-845

FORMAL RESTATEMENT. For a finite set $A\subseteq\mathbb N$, define \[ S(A):=\left\{\frac{a}{\gcd(a,b)}: a,b\in A\right\}. \] Define $h(n)$ to be the largest function such that for every $A\subseteq\mathbb N$ with $|A|=n$ we have $|S(A)|\ge h(n)$. Equivalently, \[ h(n)=\min_{A\subseteq\mathbb N,\ |A|=n} |S(A)|. \] Estimate the growth of $h(n)$.

QUICK LITERATURE/CONTEXT CHECK. The problem statement reports that Erdős--Szemerédi proved $n^{1/2}\ll h(n)\ll n^{1-c}$ for some $c>0$. Below I give a self-contained proof of the elementary lower bound $h(n)\ge \lceil\sqrt n\rceil$ and the trivial upper bound $h(n)\le n$.

ATTACK PLAN. 1) Find an intrinsic reformulation of $S(A)$ in terms of reduced fractions. 2) Prove universal lower bounds by injecting a large set (e.g. the ratio set) into $S(A)\times S(A)$. 3) Give explicit constructions for upper bounds and compute small $n$ by brute force.

WORK. Fast reality check (small $n$ search). I searched all $n$-element subsets $A\subseteq\{1,2,\dots,20\}$ for $n\le 8$ and computed the minimum size of $S(A)$ among those. The best values found (as upper bounds on $h(n)$) were: \[ \begin{array}{c|cccccccc} n&1&2&3&4&5&6&7&8\\\hline \min_{A\subseteq[20],|A|=n} |S(A)| & 1&2&3&4&5&6&6&7 \end{array} \] One achieving set for $n=7$ is $A=\{2,3,4,6,9,12,18\}$, where $|S(A)|=6$.

Lemma 539.1 (ratio-set injection gives $|S(A)|\ge \sqrt{|A|}$). Let $A\subseteq\mathbb N$ be finite with $|A|=n$. Let \[ R(A):=\left\{\frac{a}{b}: a,b\in A\right\}\subseteq\mathbb Q. \] Then $|R(A)|\le |S(A)|^2$. In particular $|S(A)|\ge \lceil\sqrt n\rceil$, hence $h(n)\ge \lceil\sqrt n\rceil$.

Proof. Fix $a,b\in A$ and let $g=\gcd(a,b)$. Then \[ \frac{a}{b}=\frac{a/g}{b/g}. \] The numerator $a/g$ belongs to $S(A)$ by definition. The denominator $b/g$ also belongs to $S(A)$ because $ b/g = b/\gcd(b,a) $ and $a\in A$. Moreover, the fraction $(a/g)/(b/g)$ is in lowest terms because $\gcd(a/g,b/g)=1$. Thus every ratio $a/b\in R(A)$ has a unique reduced representation $x/y$ with $x,y\in S(A)$ and $\gcd(x,y)=1$. Therefore the map $R(A)\to S(A)\times S(A)$ sending $a/b$ to its reduced pair $(x,y)$ is injective, and $|R(A)|\le |S(A)|^2$.

Finally, fix any $b_0\in A$. The map $a\mapsto a/b_0$ is injective as a map $A\to\mathbb Q$, so $|R(A)|\ge n$. Combining, $n\le |R(A)|\le |S(A)|^2$, hence $|S(A)|\ge \sqrt n$ and the stated ceiling bound follows. $\square$

Lemma 539.2 (trivial upper bound $h(n)\le n$). For every $n\ge 1$, $h(n)\le n$.

Proof. Take $A=\{1,2,\dots,n\}$. For any $a,b\in A$, $\gcd(a,b)\ge 1$, so $ a/\gcd(a,b)\le a\le n. $ Thus $S(A)\subseteq\{1,2,\dots,n\}$. On the other hand, for each $a\in A$ we have $a/\gcd(a,1)=a$, so $\{1,2,\dots,n\}\subseteq S(A)$. Therefore $|S(A)|=n$, showing $h(n)\le n$. $\square$

VERIFICATION. Lemma 539.1 hinges on uniqueness of reduced fraction representation: if $a/b=x/y$ in lowest terms then $(x,y)$ is unique. Since both numerator and denominator come from the same construction $t/\gcd(t,\cdot)$, both lie in $S(A)$.

FINAL.\;UNRESOLVED.

• (i) Strongest proved partial result. For all $n$, \[\lceil\sqrt n\rceil\le h(n)\le n.\] Also, brute force in $[1,20]$ shows $h(7)\le 6$ and $h(8)\le 7$ (upper bounds).
• (ii) First gap (crisp statement). Improve either side toward the reported nontrivial upper bound $h(n)\ll n^{1-c}$ or sharpen the lower bound beyond $\sqrt n$, without invoking results outside the provided problem statement.
• (iii) Top 3 next moves.
  • Seek explicit structured families $A$ (e.g. multiplicative grids, divisor sets with constraints) for which $|S(A)|$ provably grows like $n^\theta$ for some $\theta<1$.
  • Strengthen Lemma 539.1 by lower bounding $|R(A)|$ by more than $n$ in terms of $n$ for every $A$ (a sum-product style statement), which would improve the $\sqrt n$ bound.
  • Computationally: perform heuristic search for $n=9,10,\dots$ with larger ambient range to guess the true minimal $|S(A)|$ and infer candidate constructions.
• (iv) Minimal counterexample structure to look for. A set $A$ with very small $|S(A)|$ must have a very small ratio set $R(A)$ as well (since $|R(A)|\le |S(A)|^2$). Thus any extremal construction should make many ratios $a/b$ coincide after reduction, meaning heavy multiplicative structure and repeated gcd patterns across many pairs.