Published August 15, 2026
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Ramsey Number $R(4, 20) \ge 252$
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Description
We exhibit an explicit circulant graph of prime order $251$ that are $K_4$-free and have independence number $19$. Consequently
\[R(4,20)\ge 252.\]
These improve the bound $R(4,20)\ge 237$ given by Nagda, Raghavan, Thakurta and the long standing bound $R(4,21)\ge 242$ recorded in Radziszowski's dynamic survey. The graphs are $32$-subsets of a pair of undirected quintic cyclotomic classes modulo $251$, in analogy with
the quartic-residue circulant of order $313$ used for $R(4,22)$. Clique-freeness is elementary; the independence-number claims are certified by a bitset branch-and-bound on the $186$-vertex residual of a vertex.
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Additional details
References
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