Ramsey linear families and generalized subdivided graphs

The following results are obtained. (i) Let p, d, and k be fixed positive integers, and let G be a graph whose vertex set can be partitioned into parts V1, V2,…, Va such that for each i at most d vertices in V1 ∪ … ∪ Vi have neighbors in Vi+1 and r(Kk, 〈Vi〉) ⩽ p | V(G) |, where 〈Vi〉 denotes the subgraph of G induced by Vi. Then there exists a number c depending only on p, d, and k such that r(Kk, G)⩽c | V(G) |. (ii) Let d be a positive integer and let G be a graph in which there is an independent set I ⊆ V(G) such that each component of G−I has at most d vertices and at most two neighbors in I. Then r(G,G)⩽c | V(G) |, where c is a number depending only on d. As a special case, r(G, G) ⩽ 6 | V(G) | for a graph G in which all vertices of degree at least three are independent. The constant 6 cannot be replaced by one less than 4.