Ramsey Numbers for Theta Graphs

The graph Ramsey number (1,2) is the smallest integer with the property that any complete graph of at least vertices whose edges are colored with two colors (say, red and blue) contains either a subgraph isomorphic to 1 all of whose edges are red or a subgraph isomorphic to 2 all of whose edges are blue. In this paper, we consider the Ramsey numbers for theta graphs. We determine (4,), (5,) for ≥4. More specifically, we establish that (4,)=(5,)=2？1 for ≥7. Furthermore, we determine (,) for ≥5. In fact, we establish that (,)=(3/2)？1 if is even, 2？1 if is odd.