RAMSEY SIZE-LINEAR GRAPHS AND RELATED QUESTIONS.

In this paper we prove several results on Ramsey numbers R(H,F) for a fixed graph H and a large graph F, in particular for F = Kn. These results extend earlier work of Erd\H os, Faudree, Rousseau, and Schelp and of Balister, Schelp, and Simonovits on so-called Ramsey sizelinear graphs. Among other results, we show that if H is a subdivision of K4 with at least six vertices, then R(H,F) =O(v(F)+e(F)) for every graph F. We also conjecture that if H is a connected graph with e(H) v(H) (k+1/2)2, then R(H,Kn) = O(nk). The case k = 2 was proved by Erd\H os, Faudree, Rousseau, and Schelp. We prove the case k = 3. [ABSTRACT FROM AUTHOR]