The connected Ramsey number

AbstractA graph G is totally connected if both G and Ḡ (its complement) are connected. The connected Ramsey number rc(F, H) is the smallest integer k ⩾ 4 so that if G is a totally connected graph of order k then either F ⊂ G or H ⊂ Ḡ. We show that if neither of F nor H contains a bridge, then rc = r(F, H), the usual generalized Ramsey number of F and H. We compute rc (Pm Pm), the connected Ramsey number for paths