6.2 Schur numbers and their properties

Schur numbers, named after Issai Schur, are crucial in Ramsey theory. They represent the largest integer where a set can be split into sum-free subsets. Known values include S(1) = 1, S(2) = 4, S(3) = 13, and S(4) = 44.

Schur numbers grow at least exponentially, with lower and upper bounds established. Their exact growth rate remains unknown, with S(5) still unsolved. These numbers have applications in number theory and additive combinatorics, particularly in studying sum-free set distributions.

Schur Numbers: Definition and Calculations

Definition of Schur numbers

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• Schur numbers named after Issai Schur denoted as $S(k)$ for positive integers $k$
• Play crucial role in Ramsey theory relating to colorings of integers and connecting to sum-free sets
• $S(k)$ represents largest integer $n$ where set $\{1, 2, ..., n\}$ can be partitioned into $k$ sum-free subsets
• Sum-free set contains integers where no two elements sum to third element within set (3, 5, 8)

Calculation for small values

• Known Schur numbers: $S(1) = 1$, $S(2) = 4$, $S(3) = 13$, $S(4) = 44$
• Calculation employs exhaustive search for small values and computer-assisted proofs for larger values
• $S(2) = 4$ example: $\{1, 4\}$ and $\{2, 3\}$ form two sum-free subsets
• $S(3) = 13$ partitioned into three sum-free subsets: $\{1, 4, 10, 13\}$, $\{2, 3, 11, 12\}$, and $\{5, 6, 7, 8, 9\}$

Bounds and Growth of Schur Numbers

Bounds of Schur numbers

• Lower bounds: $S(k) \geq 3^k - 1$ proved by Abbott and Moser
• Upper bounds: $S(k) < k!e^k$ derived from Ramsey's theorem
• Lower bound closer to actual values for small $k$ (S(3) = 13, lower bound = 8)
• Upper bound becomes more relevant for larger $k$ (S(4) = 44, upper bound ≈ 261)

Growth rate analysis

• Asymptotic behavior shows $S(k)$ grows at least exponentially with $k$
• Grows faster than diagonal Ramsey numbers but slower than certain off-diagonal Ramsey numbers
• Exact growth rate remains unknown, determining $S(5)$ major unsolved problem
• Applications extend to number theory and additive combinatorics (sum-free set distributions)