On Some Open Questions for Ramsey and Folkman Numbers

Abstract

We discuss some of our favorite open questions about Ramsey numbers and a related problem on edge Folkman numbers. For the classical two-color Ramsey numbers, we first focus on constructive bounds for the difference between consecutive Ramsey numbers. We present the history of progress on the Ramsey number R(5, 5) and discuss the conjecture that it is equal to 43. For the multicolor Ramsey numbers, we focus on the growth of R r (k), in particular for k = 3. Two concrete conjectured cases, R(3, 3, 3, 3) = 51 and R(3, 3, 4) = 30, are discussed in some detail. For Folkman numbers, we present the history, recent developments, and potential future progress on F e (3, 3; 4), defined as the smallest number of vertices in any K4-free graph which is not a union of two triangle-free graphs. Although several problems discussed in this paper are concerned with concrete cases and some involve significant computational approaches, there are interesting and important theoretical questions behind each of them.