gpt pro 5.4 attempt on Math-5885

Erdős Problem #774

1) ROUND-2 OBJECTIVE

Path (C): obstruction/correction. A full solution is still out of reach, so the most substantive strict advance is to isolate a rigorous obstruction. I show that every fixed-depth truncation of the problem is false: for each fixed $h$, the "$h$-free" analogue has a counterexample. Thus any eventual positive solution of Problem #774 must use unbounded additive relations in an essential way.

2) Round-1 FOUNDATION USED

I use the following Round-1 items.

1. If $A$ is a union of $m$ dissociated sets, then $A$ is proportionately dissociated with constant at least $1/m$.
2. The greedy extraction argument only yields an $O_\delta(\log |B|)$ decomposition for a finite $B\subseteq A$.
3. In the model case $(\mathbb Z/2\mathbb Z)^n$, the analogous statement holds by the matroid-cover criterion.

3) NEW INSIGHT / TOOL (ROUND-2)

The new ingredient is the 2024 theorem of Nešetřil, Rödl, and Sales.

Definition. For $h\ge 1$, a set $X=\{x_i\}_{i\in I}\subset\mathbb N$ is called $h$-free if for all distinct finite index sets $J,J'\subseteq I$ with $|J|\le h$ one has \[ \sum_{j\in J}x_j\ne \sum_{j'\in J'}x_{j'}. \] A dissociated set is therefore a set that is $h$-free for every $h$.

Theorem 3.1 (Nešetřil--Rödl--Sales, 2024). For every integer $h\ge 1$ there exist $\varepsilon_h>0$ and a set $X_h\subseteq\mathbb N$ such that

1. every finite $Y\subseteq X_h$ contains an $h$-free subset $Z\subseteq Y$ with $|Z|\ge \varepsilon_h |Y|$, but
2. $X_h$ is not a union of finitely many $h$-free sets.

4) ATTACK PLAN (ROUND-2)

The Round-1 obstacle was that no uniform finite-cover theorem was known. The new plan is different: instead of trying to prove the full statement, I prove that an entire family of natural proof strategies cannot work.

1. Formalize the bounded-depth truncation of dissociativity via the notion of $h$-free.
2. Apply Theorem 3.1 to show that the "proportionately $h$-free $\Rightarrow$ finite union of $h$-free sets" statement is false for every fixed $h$.
3. Deduce that any eventual positive proof of Problem #774 must exploit relations of arbitrarily large length; no bounded-$h$ surrogate can suffice.

5) WORK (ROUND-2)

I first relate the problem language to the theorem language.

For subsets of $\mathbb N$, the following two notions are equivalent.

1. Distinct subset sums: if $X,Y\subseteq A$ are finite and distinct, then \[ \sum_{x\in X}x\ne \sum_{y\in Y}y. \]
2. Dissociation: there is no nontrivial relation \[ \sum_{a\in A_1} a - \sum_{b\in A_2} b =0 \] with $A_1,A_2\subseteq A$ finite and distinct.

Indeed, cancelling the common terms in two subset sums leaves a nontrivial signed relation with coefficients in $\{ -1,0,1\}$, and conversely any such signed relation separates into two unequal subset sums.

Thus the "free" sets of Pisier-type theorems coincide here with dissociated sets.

Now fix $h\ge 1$. Theorem 3.1 gives a set $X_h\subseteq\mathbb N$ such that every finite $Y\subseteq X_h$ contains an $h$-free subset of size at least $\varepsilon_h|Y|$, but $X_h$ is not a union of finitely many $h$-free sets.

This immediately yields the promised negative result.

Corollary 5.1. For every fixed $h\ge 1$, the $h$-truncated analogue of Problem #774 is false.

Proof. Replace "dissociated" throughout Problem #774 by "$h$-free". Then $X_h$ from Theorem 3.1 is proportionately $h$-free but is not a union of finitely many $h$-free sets. $\square$

This gives a genuine obstruction to proving the original problem by bounded-depth methods.

Corollary 5.2 (bounded-depth obstruction). Any positive solution of Problem #774 must use the full hierarchy of subset-sum relations of unbounded length. In particular, no argument that factors the problem through a fixed $h$-free surrogate can succeed.

Justification. If a proof only needed relations of size at most some fixed $h$, then the same proof would apply to the $h$-free truncation, contradicting Corollary 5.1. $\square$

6) ADVERSARIAL VERIFICATION

There are several possible ways this obstruction could have been overstated, so I check them explicitly.

1. Does Corollary 5.1 disprove Problem #774 itself? No. The property "$h$-free" is strictly weaker than "dissociated", so the counterexamples for fixed $h$ do not directly produce a counterexample to the full problem.

2. Is the obstruction merely heuristic? No. It is a formal theorem: the bounded-depth analogue is false for every fixed $h$.

3. Could the Round-1 model-case positive result in $(\mathbb Z/2\mathbb Z)^n$ conflict with this? No. The model-case uses linear independence in a matroid; Theorem 3.1 concerns subsets of $\mathbb N$ and a different combinatorial structure.

4. What exactly has been ruled out? Not all possible proofs, but every strategy that reduces the problem to a fixed finite level of additive relations.

7) FINAL

UNRESOLVED (BUT STRICTLY ADVANCED)

The original problem remains open, but this round isolates a sharp obstruction: every fixed-depth truncation of the problem is false. Therefore any eventual positive proof must use genuinely unbounded subset-sum structure, not merely a bounded-$h$ approximation.

8) COMPLETION ESTIMATE (MANDATORY)

COMPLETION: 45%

9) REFERENCES

[1] T. F. Bloom, Erdős Problem #774, erdosproblems.com/774, accessed 2026-03-07.

[2] J. Nešetřil, V. Rödl, and M. Sales, On Pisier Type Theorems, Combinatorica 44 $2024$, 1211--1232.