gpt pro 5.4 attempt on Math-924

Problem 757 (Round 2)

1) ROUND-2 OBJECTIVE

I pursue (C) obstruction/correction. The Round-1 writeup isolated the right local obstruction --- repeated distances collapse to $3$-term arithmetic progressions --- but it did not exploit the full hypergraph structure now available in the literature. The most promising Round-2 step is therefore to replace the Round-1 qualitative picture by a sharp quantitative framework: the A.P.-hypergraph reformulation, the best presently verified lower bound $c_*\ge 9/17$, and the explicit upper bound $c_*\le 4/7$.

2) Round-1 FOUNDATION USED

I use the following Round-1 results without reproving them.

• Lemma 4.1 (Round 1). In a $(4,5)$-set, two disjoint pairs cannot determine the same distance.
• Corollary 4.2 (Round 1). For $S\subseteq A$, the subset $S$ is Sidon if and only if $S$ contains no nontrivial $3$-term arithmetic progression.

These are exactly the ingredients needed to pass from additive combinatorics to a $3$-uniform hypergraph independence problem.

3) NEW INSIGHT / TOOL (ROUND-2)

The new ingredient is the A.P.-hypergraph of a $(4,5)$-set. For a finite $A\subset\mathbb R$, define $H(A)$ to be the $3$-uniform hypergraph with vertex set $A$ and edge set \[ E(H(A)):=\left\{\{x-d,x,x+d\}\subseteq A: d\neq 0\right\}. \] By Round-1 Corollary 4.2, the size of the largest Sidon subset of $A$ is precisely the independence number of $H(A)$.

Round 2 adds three structural facts.

1. (i) $H(A)$ is linear: distinct edges intersect in at most one vertex.
2. (ii) Distinct edges of $H(A)$ have distinct midpoints; consequently $|E(H(A))|\le |A|-2$.
3. (iii) (Gyárfás--Lehel) If $A$ is a $(4,5)$-set, then $H(A)$ is $F_7$-free, where $F_7$ is the $7$-vertex $3$-uniform hypergraph with edges \[ \{0,1,2\},\ \{0,3,4\},\ \{0,5,6\},\ \{1,3,5\},\ \{2,4,6\}. \] Then the Henning--Yeo transversal theorem for $3$-uniform linear $F_7$-free hypergraphs gives the improved lower bound $c_*\ge 9/17$.

A second new ingredient is an explicit $14$-point $(4,5)$-set with largest Sidon subset of size $8$, which gives the improved upper bound $c_*\le 4/7$.

4) ATTACK PLAN (ROUND-2)

The exact gap after Round 1 was quantitative:

• Round 1 reduced the problem to finding a large $3$-term-A.P.-free subset.
• What remained was a numerical lower bound on the largest independent set in the associated hypergraph, and a better construction for the upper bound.

I therefore prove the hypergraph translation cleanly, verify the linearity and midpoint-counting properties directly from the Round-1 lemmas, and then import the two external facts that Round 1 did not use:

1. the Gyárfás--Lehel $F_7$-freeness theorem, and
2. the Henning--Yeo bound $17\tau(H)\le 5|V(H)|+3|E(H)|$ for $3$-uniform linear $F_7$-free hypergraphs.

This closes the main Round-1 quantitative gap.

5) WORK (ROUND-2)

For a $(4,5)$-set $A$, let $h(A)$ denote the maximum size of a Sidon subset of $A$. For each $n\ge 1$, let \[ \sigma(n):=\min\{h(A): A\subset\mathbb R,\ |A|=n,\ A\text{ is a }(4,5)\text{-set}\}, \] and let \[ c_*:=\inf_{n\ge 1}\frac{\sigma(n)}{n}. \]

Proposition 5.1.

For every $(4,5)$-set $A\subset\mathbb R$, \[ h(A)=\alpha(H(A)), \] where $\alpha$ denotes the independence number.

Proof. By Round-1 Corollary 4.2, a subset $S\subseteq A$ is Sidon if and only if it contains no nontrivial $3$-term arithmetic progression. But the edges of $H(A)$ are exactly the nontrivial $3$-term arithmetic progressions contained in $A$. Hence $S$ is Sidon if and only if $S$ is an independent set of $H(A)$. Taking the maximum over all $S\subseteq A$ yields $h(A)=\alpha(H(A))$. $\square$

Proposition 5.2.

If $A$ is a $(4,5)$-set, then $H(A)$ is linear.

Proof. Suppose two distinct edges of $H(A)$ intersect in at least two vertices. Since every edge has size $3$, the two edges then share exactly two vertices. Let the shared vertices be $x<y$, and put $d:=y-x>0$.

Any nontrivial $3$-term arithmetic progression containing both $x$ and $y$ has third point equal to one of \[ x-d,\qquad \frac{x+y}{2},\qquad y+d. \] Since the two edges are distinct, their union is obtained by choosing two distinct points from this three-element set. After translating by $-x$ and scaling by $d$, the resulting four-point set is one of \[ \{-1,0,1,2\},\qquad \{-1,0,\tfrac12,1\},\qquad \{0,\tfrac12,1,2\}. \] In the first case the positive distances are $\{1,2,3\}$, and in each of the other two cases the positive distances are $\{\tfrac12,1,\tfrac32,2\}$. Thus in every case the $4$-point union has at most four distinct positive distances, so its difference set has size at most \[ 1+2\cdot 4=9, \] contradicting the $(4,5)$-condition $|B-B|\ge 11$ for every $4$-subset $B\subseteq A$.

Therefore two distinct edges cannot intersect in two vertices, so $H(A)$ is linear. $\square$

Proposition 5.3.

If $A$ is a $(4,5)$-set and $|A|\ge 2$, then distinct edges of $H(A)$ have distinct midpoints. Consequently, \[ |E(H(A))|\le |A|-2. \]

Proof. Suppose two distinct edges have the same midpoint $u\in A$. Then for some distinct positive $d,e$, \[ \{u-d,u,u+d\},\ \{u-e,u,u+e\}\in E(H(A)). \] Without loss of generality $0<d<e$. Consider the $4$-set \[ B=\{u-e,u-d,u+d,u+e\}. \] Its six pairwise distances are \[ e-d,\ e+d,\ 2e,\ 2d,\ e+d,\ e-d. \] Thus among the positive distances there are at most four distinct values, namely \[ e-d,\ 2d,\ e+d,\ 2e. \] Hence \[ |B-B|\le 1+2\cdot 4=9, \] contradicting the $(4,5)$-condition $|B-B|\ge 11$. Therefore distinct edges have distinct midpoints.

Now the midpoint of any nontrivial $3$-term progression lies strictly between its two endpoints, so it cannot be the minimum or maximum element of $A$. Thus the midpoint map $E(H(A))\to A$ is injective and lands in $A\setminus\{\min A,\max A\}$. Therefore \[ |E(H(A))|\le |A|-2. \] $\square$

External theorem used.

Gyárfás and Lehel proved that if $A$ is a $(4,5)$-set, then $H(A)$ contains no copy of $F_7$.

External theorem used.

(Henning--Yeo.) If $H$ is a $3$-uniform linear hypergraph with no subhypergraph isomorphic to $F_7$, then \[ 17\tau(H)\le 5|V(H)|+3|E(H)|, \] where $\tau(H)$ is the transversal number.

Theorem 5.4.

For every $(4,5)$-set $A$ of size $n$, \[ h(A)\ge \frac{9}{17}n. \] Consequently, \[ c_*\ge \frac{9}{17}. \]

Proof. If $n=1$, then $h(A)=1\ge 9/17$, so there is nothing to prove. Assume $n\ge 2$, and let $H:=H(A)$. By Proposition 5.1, $h(A)=\alpha(H)$. By Proposition 5.2, $H$ is linear; by the Gyárfás--Lehel theorem, $H$ is $F_7$-free; and by Proposition 5.3, \[ |E(H)|\le |V(H)|-2=n-2. \] Applying the Henning--Yeo theorem gives \[ 17\tau(H)\le 5n+3(n-2)=8n-6<8n. \] Hence $\tau(H)\le 8n/17$. Since $\alpha(H)+\tau(H)=|V(H)|=n$ for every hypergraph, \[ h(A)=\alpha(H)\ge n-\frac{8n}{17}=\frac{9n}{17}. \] Taking the minimum over all $(4,5)$-sets $A$ of size $n$ yields $\sigma(n)\ge 9n/17$ for every $n$, and hence $c_*\ge 9/17$. $\square$

Theorem 5.5.

There exists a $(4,5)$-set \[ A_{\mathrm{base}}=\{0,136,200,243,246,249,272,286,298,323,400,528,596,1056\} \] of size $14$ such that $h(A_{\mathrm{base}})=8$. Consequently, \[ c_*\le \frac{8}{14}=\frac47. \]

Proof. The assertions that $A_{\mathrm{base}}$ is a $(4,5)$-set and that its largest Sidon subset has size $8$ were verified by exhaustive exact computation in Ma--Tang. Therefore $\sigma(14)\le 8$, and by the definition of $c_*$ as an infimum of the ratios $\sigma(n)/n$, we obtain $c_*\le 8/14=4/7$. $\square$

Round-2 conclusion for Problem 757.

Combining Theorems 5.4 and 5.5 gives the strongest explicit interval reached in this round: \[ \frac{9}{17}\le c_*\le \frac47. \] This is a strict improvement over the Round-1 state $\tfrac12<c_*<\tfrac35$.

6) ADVERSARIAL VERIFICATION

• Does Proposition 5.1 use anything stronger than Round 1? No. It is exactly Round-1 Corollary 4.2 rewritten hypergraphically.
• Could Proposition 5.2 fail when two $3$-term progressions share two vertices? No. The proof reduces the union to one of three affine normal forms, and each normal form has at most four distinct positive distances on its four vertices.
• Is the midpoint-counting argument valid for all edges? Yes. The midpoint of a nontrivial progression is never an extreme point of $A$, so the midpoint map lands in $A\setminus\{\min A,\max A\}$.
• Does the upper bound really follow from one finite example? Yes. Since $c_*=\inf_n \sigma(n)/n$, one example with $\sigma(14)\le 8$ immediately implies $c_*\le 8/14$.
• Is anything claimed beyond the cited external theorems? No. The only imported ingredients are the $F_7$-freeness theorem of Gyárfás--Lehel, the Henning--Yeo transversal bound, and the explicit $14$-point construction of Ma--Tang.

7) FINAL

UNRESOLVED (BUT STRICTLY ADVANCED). Round 2 upgrades the Round-1 qualitative reduction to the presently strongest quantitative statement I could verify: \[ \frac{9}{17}\le c_*\le \frac47. \] The lower bound is now derived by a clean hypergraph argument using Round-1's "Sidon $\Leftrightarrow$ $3$AP-free" reformulation, while the upper bound comes from an explicit $14$-point example.

8) COMPLETION ESTIMATE (MANDATORY)

COMPLETION: 55%

9) REFERENCES

• J. Ma and Q. Tang, Largest Sidon subsets in weak Sidon sets, arXiv:2602.23282, 2026.
• A. Gyárfás and J. Lehel, Linear sets with five distinct differences among any four elements, J. Combin. Theory Ser. B 64 $1995$, 108--118.
• M. A. Henning and A. Yeo, Affine Planes and Transversals in 3-Uniform Linear Hypergraphs, Graphs Combin. 37 $2021$, 867--890.

10) OUTPUT FORMAT

Here the Erdős problem number is $x=757$.