# Erdős Open Prize Problem Solver

## Problem {problemId}
{problemText}

{commentsText}

## Instructions

You are solving an open Erdős prize problem. These are long-standing unsolved problems — most have resisted expert attack for decades. A rigorous partial result or a well-documented failure is more valuable than a fake proof.

### What to do

**1. Prior work and barriers** (brief — 1-2 paragraphs)
State the best known bounds or partial results with attribution. Identify why the problem remains open: what specifically fails when you try standard approaches? If you recognize the problem, say what is known. Do NOT pad this section with textbook definitions.

**2. Attack** (the bulk of your response)
Attempt a proof, disproof, or partial result. Requirements:
- Number all major logical steps (Step 1, Step 2, ...).
- Cite every theorem you invoke by name and verify its hypotheses apply to your setting.
- If a step requires an unproven claim, mark it with ⚠️ and state it as a conditional: "If X holds, then..."
- If you reach an impasse, document exactly where and why the argument breaks. Then try an alternative approach. Attempting 2-3 distinct approaches that each fail informatively is better than one long approach that trails off.
- If you cannot improve on known results, say so explicitly. Do not repackage known bounds as new results.

**3. Verification**
Before finalizing, re-examine your argument adversarially:
- For each key step, ask: would a skeptical referee accept this? Is any implication actually an equivalence? Are quantifiers in the right order?
- Check: do boundary cases (n=1, 2, 3) or degenerate cases break the argument?
- Check: did you use any theorem whose hypotheses you did not verify?
- Check: does your claimed result contradict any known construction or lower/upper bound?
- If you find an error, fix it or downgrade your claim. Do not leave a known error in the final output.

### Common failure modes to avoid
- **Survey masquerading as progress**: Summarizing known results without contributing anything new. If all your content is attributable to prior work, classify as No Progress.
- **Wrong asymptotic regime**: Proving O(f(n)) when the problem asks for o(f(n)), or proving an upper bound when a lower bound is needed.
- **Unverified theorem application**: Invoking a theorem (Szemerédi Regularity, Hales-Jewett, etc.) without checking that your objects satisfy its hypotheses.
- **Circular reasoning**: Using the conclusion as a hidden assumption, especially in probabilistic arguments.
- **Overclaiming**: Stating "this proves the conjecture" when what you proved is a weaker statement or a special case.

## Output Format

### Prior Work
[Best known results, key references by author name, and why the problem is open.]

### Approach
[Which strategy you chose and why. If multiple approaches were attempted, describe each.]

### Solution / Analysis
[The complete mathematical argument with numbered steps. If partial, clearly state what is proved vs. what remains open. Mark conditional steps with ⚠️.]

### Verification
[Your self-check. For each key deduction, state whether it survives scrutiny. If you found and fixed errors, document them.]

### Result
- **Status**: [Solved / Major Partial Result / Minor Partial Result / Documented Failure / No Progress]
- **What is proved**: [One sentence stating your strongest unconditional result, or "No new result beyond known bounds."]
- **Open sub-problems**: [If applicable, specific sub-problems whose resolution would complete the proof.]