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

Abstract

For any two graphs F1 and F2, the graph Ramsey number r(F1, F2) is the smallest positive integer N with the property that every graph on at least N vertices contains F1 or its complement contains F2 as a subgraph. In this paper, we consider the Ramsey numbers for theta-complete graphs. We determine r(θn, Km) for m = 2, 3, 4 and n > m. More specifically, we establish that r(θn, Km) = (n − 1)(m − 1) + 1 for m = 3, 4 and n > m.