Exact value of the diagonal Ramsey number R(5)

Determine the exact value of the diagonal Ramsey number R(5), defined as the smallest integer n such that every red–blue edge-coloring of the complete graph K_n contains either a red 5-clique or a blue 5-clique, given the current best bounds 43 ≤ R(5) ≤ 46.

Background

The paper reviews classical results on diagonal Ramsey numbers and notes the long-standing difficulty of pinning down exact values beyond small cases. While R(3)=6 and R(4)=18 are known, the next case, R(5), has resisted determination for decades.

Recent progress has narrowed the bounds to 43 ≤ R(5) ≤ 46, but the exact value remains unresolved. The author highlights this as an emblematic open problem in Ramsey theory to contextualize the significance of improved upper bounds and formal verification efforts.

References

It's known that R(4)=18 and that 43\le R(5)\le 46. Erdős joked about the difficulty of determining $R(5)$ more than 30 years ago, and we still don't know it.

— Formalising New Mathematics in Isabelle: Diagonal Ramsey  (2501.10852 - Paulson, 18 Jan 2025) in Section 2 (Ramsey's theorem), paragraph following Figure 1