Determine the exact value of the Ramsey number R(4,20), defined by R(4,20)=minn∈N: every graph on n vertices contains either a copy of K_4 or an independent set of 20 vertices.

Ramsey number R(4,20)

Determine the exact value of the Ramsey number $R(4,20)$ , defined by $R(4,20)=\min\{n\in\mathbb{N}:\text{ every graph on }n\text{ vertices contains either a copy of }K_4\text{ or an independent set of }20\text{ vertices}\}$ .

Sources & referencesView supporting material

Primary source

arXiv

Additional references

Ramsey number $R(4, 20) \ge 252$ — arXiv — Yu, Charlie

computational-combinatorics ramsey-theory

1 day ago

Progress summary

Refreshed 1 day ago

Partially solved

A new computationally certified construction raises the known lower bound from $237$ to $252$ , but the exact number remains unknown.

The problem asks for the exact finite Ramsey number $R(4,20)$ . The latest result gives a substantially stronger lower bound, but does not determine the number.

August 2026 lower-bound improvement

Yu, Charlie reports $R(4,20) \ge 252$ , improving the previous recorded lower bound $237$ . The construction is circulant and its independence-number certification is computational; the exact Ramsey number remains open.

Current status (as of August 2026): $R(4,20) \ge 252$ is reported, while no matching upper bound or exact determination is established.

Sources

arxiv.org

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