6.1 Statement and proof of Schur's Theorem

Schur's Theorem is a cornerstone of Ramsey Theory. It guarantees that in any coloring of a large enough set of integers, you'll find three numbers of the same color that add up. This simple idea has far-reaching consequences.

The proof uses clever tricks like the pigeonhole principle to show why this must be true. It's a great example of how basic concepts can lead to powerful results in mathematics, sparking new areas of research.

Schur's Theorem: Statement and Proof

Schur's Theorem in Ramsey theory

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• Schur's Theorem states for any positive integer $r$, there exists a positive integer $[S(r)](https://www.fiveableKeyTerm:s(r))$ such that partitioning $\{1, 2, ..., S(r)\}$ into $r$ parts guarantees $x$, $y$, and $z$ in the same part satisfying $x + y = z$ (monochromatic solution)
• Establishes existence of Ramsey-type numbers for additive equations demonstrating inevitability of certain structures in large enough sets (arithmetic progressions)
• Provides foundation for more general Ramsey-type theorems led to developments in extremal combinatorics and additive number theory

Proof steps of Schur's Theorem

• Proof approach uses pigeonhole principle to find monochromatic solution
• Key steps:
  1. Consider equation $x + y = z$ in $r$ colors
  2. Construct larger set $S = \{1, 2, ..., N\}$ where $N = r^{r^2} + 1$
  3. Define $T = \{(a, b) : 1 \leq a < b \leq N\}$
  4. Color pairs in $T$ based on color of $b - a$ in original coloring
  5. Apply pigeonhole principle to find monochromatic $(x, y)$, $(y, z)$, and $(x, z)$
  6. Conclude $x + (y - x) = z$ is monochromatic solution
• Demonstrates power of pigeonhole principle in Ramsey theory proofs

Applications of Schur's Theorem

• Problem-solving strategies:
  • Identify number of colors in partition (2-color, 3-color)
  • Determine set size needed for monochromatic solution (Schur numbers)
  • Find triples satisfying $x + y = z$ in same color
• Applications include analyzing integer colorings for monochromatic arithmetic progressions (van der Waerden's theorem)
• Study partitions of finite groups to find monochromatic solutions to equations (cyclic groups)
• Investigate graph colorings to find monochromatic substructures (Ramsey graphs)

Significance for Ramsey theory

• Proved by Issai Schur in 1916 predating Ramsey's theorem (1930) shared similar concepts
• Introduced idea of guaranteed structures in large enough sets sparked interest in finding Ramsey numbers for various structures (complete graphs)
• Led to generalizations in different mathematical contexts connected number theory (arithmetic progressions), combinatorics (partition regularity), and algebra (finite groups and rings)