The Ramsey number for theta graph versus a clique of order three and four

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 > m. More specifically, we establish that $r(θ_n,K_m) = (n − 1)(m − 1) + 1$ for m = 3, 4 and n > m