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Set Ramsey Number

The set Ramsey number r_k(m,n), for integers k>=2 and 1<=m,n<=(k; 2), is the least integer N such that every simple graph on N vertices has either k vertices inducing at least m edges or k vertices inducing at least n edges in its graph complement (Guo and Wan 2026). It is distinct from the set-coloring Ramsey number, which assigns sets of colors to edges.

Guo and Wan (2026) proved that r_5(10,8)=21, using the equivalent notation R(5,10;5,8). They report a computer-assisted enumeration of exactly 17 nonisomorphic 20-vertex graphs avoiding both conditions, followed by exclusion of every one-vertex extension. Guo and Wan (2026) report DeepSeek assistance with writing computational code, which the authors independently verified.

See also

Ramsey Number, Set-Coloring Ramsey Number

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References

Guo, W. and Wan, M. "A Computer-Assisted Proof that R(5,10;5,8)=21 for the Generalized Threshold Ramsey Number." 21 Oct 2026. https://arxiv.org/abs/2610.29584.

Cite this as:

Weisstein, Eric W. "Set Ramsey Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SetRamseyNumber.html

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