Mathematical Techniques Library

A curated collection of proof techniques relevant to Erdős problems, organized by problem domain.

Quick Reference

Domain	Common Techniques
Number Theory	Pell equations, quadratic forms, modular arithmetic
Combinatorics	Pigeonhole, probabilistic method, counting arguments
Graph Theory	Ramsey theory, chromatic bounds, incidence geometry
Geometry	Lattice constructions, distance sets, polynomial methods

Technique Files

For detailed technique descriptions, see:

• NUMBER_THEORY.md - Prime sequences, Diophantine equations
• COMBINATORICS.md - Counting, Ramsey, extremal
• GRAPH_THEORY.md - Coloring, distances, matchings
• GEOMETRY.md - Point sets, distances, lattices
• LITERATURE.md - Key papers and reductions

Problem-Solving Workflow

Phase 1: Problem Analysis

1. Parse the LaTeX statement carefully
2. Identify the problem domain (tags)
3. Check if formalized in Lean (lean_url field)
4. Read original Erdős paper references

Phase 2: Literature Check

1. Search for problem number in academic databases
2. Check Terence Tao's wiki for AI progress
3. Look for related OEIS sequences
4. Review comments on erdosproblems.com

Phase 3: Technique Selection

Based on problem type:

• Existence proofs: Probabilistic method, constructions
• Bounds: Incidence geometry, polynomial methods
• Sequences: Pell equations, growth rate analysis
• Graphs: Ramsey theory, chromatic number bounds

Phase 4: Solution Attempt

1. Try simplest applicable technique first
2. Look for reductions to known results
3. Check for counterexamples if conjecture
4. Verify small cases computationally

Phase 5: Verification

1. Check solution addresses intended interpretation
2. Search literature for existing solutions
3. Verify with formal tools if possible
4. Have human expert review

Success Patterns from AI Solutions

Erdős-652 (Distinct Distances)

Technique: Reduction to Pach-Sharir incidence bounds Pattern: Connect discrete geometry to incidence theory

Erdős-1051 (Irrationality of Series)

Technique: Mahler's criterion, growth rate analysis Pattern: Establish contradiction via asymptotic bounds

Erdős-654 (Four Points No Circle)

Technique: Axis-aligned construction with prime powers Pattern: Use number-theoretic constraints for geometric problems

Erdős-935 (Powerful Parts)

Technique: Pell equation solutions + Dirichlet's theorem Pattern: Construct specific sequences via Diophantine equations

Erdős-397 (Central Binomial Products)

Technique: Explicit parametric family construction Pattern: Find algebraic identity yielding infinite solutions

Warning Signs

• Problem seems "too easy" → likely solved or misinterpreted
• Solution doesn't use domain-specific tools → check interpretation
• Can't find any related literature → problem may be stated incorrectly
• Proof works for all cases trivially → definitional ambiguity likely