An exact value for the Ramsey number $R(K_5, K_{5-e})$

We compute the exact value of the Ramsey number $R(K_5, K_{5-e})$. It is equal to 30.

💡 Research Summary

The paper determines the exact value of the Ramsey number R(K₅, K₅‑e), proving that it equals 30. The authors begin by recalling the definition of Ramsey numbers: R(s, t) is the smallest integer n such that every graph on n vertices contains either a complete subgraph Kₛ or an independent set of size t. The variant R(Kₛ, Kₜ‑e) asks for a Kₛ or a set of t vertices with at most one edge among them (often denoted Kₜ·5). Prior work only bounded R(5, 4.5) between 30 and 33. To close this gap, the authors aim to show that no graph of order 30 avoids both a K₅ and a K₅‑e, i.e., the set R(5, 4.5, 30) is empty.

The proof proceeds in several stages. First, the authors compute exhaustive censuses of two auxiliary families of Ramsey graphs: R(4, 4.5) (graphs avoiding K₄ and a K₄‑e) and R(5, 3.5) (graphs avoiding K₅ and a K₃‑e). Using a simple “one‑vertex extender” algorithm, they generate all such graphs up to the necessary orders, recording the number of graphs and the range of edge counts for each order (Table 1). This step relies heavily on the Nauty package for isomorphism rejection and consumes only a few CPU hours.

Next, they introduce a linear‑programming constraint, called the “edge equation”: for any vertex v in a graph F of order n, \