Density theorems for bipartite graphs and related Ramsey-type results

In this paper, we present several density-type theorems which show how to find a copy of a sparse bipartite graph in a graph of positive density. Our results imply several new bounds for classical problems in graph Ramsey theory and improve and generalize earlier results of various researchers. The proofs combine probabilistic arguments with some combinatorial ideas. In addition, these techniques can be used to study properties of graphs with a forbidden induced subgraph, edge intersection patterns in topological graphs, and to obtain several other Ramsey-type statements. 1 Background and Introduction Ramsey theory refers to a large body of deep results in mathematics whose underlying philosophy is captured succinctly by the statement that “In a large system, complete disorder is impossible.” This is an area in which a great variety of techniques from many branches of mathematics are used and whose results are important not only to graph theory and combinatorics but also to logic, analysis, number theory, and geometry. Since the publication of the seminal paper of Ramsey [43] in 1930, this subject has grown with increasing vitality, and is currently among the most active areas in combinatorics. For a graph H, the Ramsey number r(H) is the least positive integer n such that every two-coloring of the edges of the complete graph Kn on n vertices, contains a monochromatic copy of H. Ramsey’s theorem states that r(H) exists for every graph H. A classical result of Erdős and Szekeres [20], which is a quantitative version of Ramsey’s theorem, implies that r(Kk) ≤ 22k for every positive integer k. Erdős [16] showed using probabilistic arguments that r(Kk) > 2k/2 for k > 2. Over the last sixty years, there have been several improvements on these bounds (see, e.g., [13]). However, despite efforts by various researchers, the constant factors in the above exponents remain the same. Determining or estimating Ramsey numbers is one of the central problem in combinatorics, see the book Ramsey theory [27] for details. Besides the complete graph, the next most classical topic in this area concerns the Ramsey numbers of sparse graphs, i.e., graphs with certain upper bound constraints on the degrees of the vertices. The study of these Ramsey numbers was initiated by Burr and Erdős in 1975, and this topic has since placed a central role in graph Ramsey theory. An induced subgraph is a subset of the vertices of a graph together with all edges whose both endpoints are in this subset. There are several results and conjectures which indicate that graphs which do not contain a fixed induced subgraph are highly structured. In particular, the most famous conjecture of this sort by Erdős and Hajnal [18] says that every graph G on n vertices which does not contain a fixed induced subgraph H has a clique or independent set of size a power of n. This is in ∗Department of Mathematics, Princeton, Princeton, NJ. Email: [email protected]. Research supported by an NSF Graduate Research Fellowship and a Princeton Centennial Fellowship. †Department of Mathematics, Princeton, Princeton, NJ. Email: [email protected]. Research supported in part by NSF CAREER award DMS-0546523, NSF grant DMS-0355497 and by a USA-Israeli BSF grant.