Geodesic
The Ramsey number for theta graph versus a clique of order three and four
Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 2, pp. 223-232

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For any two graphs F_1 and F_2, the graph Ramsey number r(F_1, F_2) is the smallest positive integer N with the property that every graph on at least N vertices contains F_1 or its complement contains F_2 as a subgraph. In this paper, we consider the Ramsey numbers for theta-complete graphs. We determine r(θ_n,K_m) for m = 2, 3, 4 and n gt; m. More specifically, we establish that r(θ_n,K_m) = (n − 1)(m − 1) + 1 for m = 3, 4 and n gt; m
Keywords: Ramsey number, independent set, theta graph, complete graph 
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Bataineh, M.S.A.; Jaradat, M.M.M.; Bateeha, M.S. The Ramsey number for theta graph versus a clique of order three and four. Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 2, pp. 223-232. http://geodesic.mathdoc.fr/item/DMGT_2014_34_2_a1/