gpt pro 5.2 attempt on Math-924

Problem 757

1) FORMAL RESTATEMENT

Let $A\subset\mathbb{R}$ with $|A|=n$. For a finite set $B\subset\mathbb{R}$, write \[ B-B:=\{x-y: x,y\in B\}. \] Assume that $A$ satisfies the $(4,5)$-difference condition: \[ \forall B\subseteq A\ (|B|=4\implies |B-B|\ge 11). \] Equivalently, for every $4$-set $B=\{b_1,b_2,b_3,b_4\}$, among the $6$ pairwise distances $\{|b_i-b_j|:1\le i<j\le 4\}$ at least $5$ are distinct.

A subset $S\subset\mathbb{R}$ is a Sidon set (a $B_2$-set) if all sums $s_i+s_j$ with $s_i\le s_j$ are distinct; equivalently, all nonzero differences $s_i-s_j$ with $s_i>s_j$ are distinct.

Define, for each $n$, the quantity \[ \sigma(n):=\min\Big\{\max\{|S|: S\subseteq A\ \text{Sidon}\}: A\subset\mathbb{R},\ |A|=n,\ A\ \text{satisfies }|B-B|\ge 11\ \forall B\in\binom{A}{4}\Big\}. \] The problem asks for the best constant \[ c^*:=\sup\{c>0: \forall n\ \sigma(n)\ge cn\}, \] (or, equivalently, the asymptotically best guaranteed linear fraction of a Sidon subset in every such $A$).

2) QUICK LITERATURE/CONTEXT CHECK

The statement as recorded in Erdős's problem list (and repeated in later sources) includes:

• Erdős--Sós: $c^*\ge \tfrac12$.
• Gyárfás--Lehel $1995$: improved the lower bound and exhibited an upper bound, \[\tfrac12<c^*<\tfrac35,\] with the upper bound given by taking $A$ to be the first $n$ Fibonacci numbers.

I did not locate a definitive resolution of the exact value of $c^*$.

3) ATTACK PLAN

A natural first step is to understand what the constraint $|B-B|\ge 11$ forbids. For a $4$-set, $|B-B|=1+2\cdot\#\{\text{distinct positive distances in }B\}$. Thus $|B-B|\ge 11$ means at most one of the $6$ distances repeats. The extremal obstruction to being Sidon is an equality of distances \[ |a-b|=|c-d|\quad\text{with }\{a,b\}\ne\{c,d\}. \] Under the $(4,5)$ hypothesis, such equalities are forced to have strong overlap structure. If one can reduce Sidon-ness in $A$ to forbidding only a small class of local configurations (e.g. $3$-term arithmetic progressions), then the problem becomes a structured extremal/packing question (hypergraph independent set problem) on those configurations.

4) WORK

Lemma 4.1 (no repeated distance on disjoint pairs).

Assume $A$ satisfies the $(4,5)$-difference condition. Then there do not exist four distinct $a,b,c,d\in A$ such that \[ |a-b|=|c-d|\ \text{ and }\ \{a,b\}\cap\{c,d\}=\varnothing. \] Proof. Assume (for contradiction) that $a,b,c,d$ are distinct and $|a-b|=|c-d|=:r$ with disjoint pairs. By translating and relabelling if needed, we may assume $a<b$ and $c<d$ so that $b-a=d-c=r$. Then also \[ |c-a|=c-a = (c+r)-(a+r)=d-b=|d-b|. \] Hence in the $4$-set $\{a,b,c,d\}$ the distance $r$ occurs at least twice (between $a,b$ and $c,d$) and the distance $|c-a|$ occurs at least twice (between $a,c$ and $b,d$), so there are at least two repeated distances among the $6$ pairwise distances. This contradicts the $(4,5)$-difference condition. $\square$

Corollary 4.2 (Sidon versus 3-term arithmetic progressions inside $A$).

Let $S\subseteq A$. Then $S$ fails to be Sidon if and only if $S$ contains a (nontrivial) $3$-term arithmetic progression, i.e. distinct $x,y,z\in S$ with $y-x=z-y$.

Proof. If $S$ contains a $3$-term AP $x<y<z$ with $y-x=z-y$, then the distance $y-x$ is repeated among pairs $(x,y)$ and $(y,z)$, so $S$ is not Sidon. Conversely, if $S$ is not Sidon, there exist distinct unordered pairs $\{u,v\}\ne\{u',v'\}$ in $S$ with $|u-v|=|u'-v'|$. By Lemma 4.1 these pairs cannot be disjoint, hence they share a vertex. Up to relabelling, we have distinct $x,y,z\in S$ with $|x-y|=|y-z|$, which in $\mathbb{R}$ implies $y$ is the midpoint of $x$ and $z$, i.e. $x,y,z$ form a $3$-term AP. $\square$

Interpretation.

Under the $(4,5)$-difference condition, the entire Sidon obstruction collapses to the presence of $3$-term arithmetic progressions. In other words, within such an $A$, finding a large Sidon subset is equivalent to finding a large subset free of $3$-term APs.

Small-$n$ sanity check (computational).

A brute-force search over small integer sets (restricted to a bounded range) finds examples of $(4,5)$-sets and computes the smallest possible maximum Sidon subset size among them. For instance, within a small search window one can find:

• $n=4$: example $A=\{0,1,2,5\}$ satisfies the condition; its largest Sidon subset has size $3$.
• $n=6$: example $A=\{0,1,2,5,8,15\}$ satisfies the condition; its largest Sidon subset has size $4$.

These are consistent with a linear guarantee, but of course do not address the optimal constant $c^*$.

5) OBSTRUCTIONS/CAVEATS

• Determining $c^*$ requires a sharp extremal analysis of how many $3$-term APs can be forced by the $(4,5)$ condition and how large an AP-free subset must exist in such structured sets.
• While Lemma 4.1 and Corollary 4.2 reduce the nature of the obstruction, they do not by themselves yield the best constant (currently only the interval $\tfrac12<c^*<\tfrac35$ is recorded).

6) FINAL

UNRESOLVED.

• (i) Where I got stuck: I did not obtain a new argument pinning down the exact optimal constant $c^*$, or improving the best known bounds beyond $\tfrac12<c^*<\tfrac35$.
• (ii) Partial progress: I proved that the $(4,5)$ condition forbids equal distances on disjoint pairs (Lemma 4.1) and therefore, within such sets, Sidon subsets are exactly those avoiding $3$-term APs (Cor. 4.2).
• (iii) What seems next: Model the family of $3$-term APs in $A$ as a $3$-uniform hypergraph with strong linearity-type constraints; then apply (or develop) sharp bounds on the independence number under those constraints. This is plausibly where the improvements from $1/2$ to $>1/2$ arise.
• (iv) What I believe is true: The exact value of $c^*$ is likely nontrivial; the Fibonacci example suggests the extremal configurations are highly structured rather than random.

7) COMPLETION ESTIMATE

$20\%$.