The CO-irredundant Ramsey number t(4,7)

Abstract

The CO-irredundant Ramsey number t(l,m) is the smallest integer n such that for any 2-edge colouring (R,B ) of K_n, the spanning subgraph B of K_n has a CO-irredundant set of size l or the subgraphR has a CO-irredundant set of size m. The 2-edge colouring (R,B) is a t(l,m) Ramsey colouring of K_n if B (R, respectively) does not contain a CO-irredundant set of size l (m, respectively); in this case R is also called an (l,m,n) Ramsey graph. It is known that t(3,m) = m for all m ≥ 3, t(4,4) = 6, t(4,5) = 8 and t(4,6) = 11 (Cockayne, MacGillivray and Simmons). We determine all (4,4,5), (4,5,7) and (4,6,10) graphs and use these to construct (4,7,13) graphs. We also show that t(4,7) ≤ 14, thus proving that t(4,7) = 14.