Abstract:

We give a general construction of a triangle free graph on 4p points whose complement does not contain K_p+2  e for p >= 4. This implies the the Ramsey number R(K_3, K_k  e) >= 4k  7 for k >= 6. We also present a cyclic triangle free graph on 30 points whose complement does not contain K_9  e. The first construction gives lower bounds equal to the exact values of the corresponding Ramsey number for k = 6, 7 and 8. the upper bounds are obtained by using computer algorithms. In particular, we obtain two new values of Ramsey numbers R(K_3, K_8  e) = 25 and R(K_3, K_9  e) = 31, the bounds 36 <= R(K_3, K_10  e) <= 39, and the uniqueness of extremal graphs for Ramsey numbers R(K_3, K_6  e) and R(K_3, K_7  e). 