A note on Ramsey multiplicity

If G₁ and G₂ are graphs and the Ramsey number r(G₁, G₂) = p, then the fewest number of G₁ in G and G₂ in ? (G complement) that occur in a graph G on p points is called the Ramsey multiplicity and denoted R(G₁, G₂). In [2, 3] the diagonal (i.e. G₁ = G₂) Ramsey multiplicities are derived for graphs on 3 and 4 points, with the exception of K₄. In this note an upper bound is established for R(K_s, K₁). Specifically, we show that R(K₄, K₄) ? 12.