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).
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Computer Science (GCCIS)
Radziszowski, Stanislaw, "The Ramsey numbers R(K_3, K_8 - e) and R(K_3, K_9 - e)" (1990). The Charles Babbage Research Centre; The Journal of Combinatorial Mathematics and Combinatorial Computing, vol. 8 (), pps. 137 - 145. Accessed from
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