We provide an elementary proof of the fact that the ramsey number of every bipartite graph H with maximum degree at most is less than 8(8) jV (H)j. This improves an old upper bound on the ramsey number of the n-cube due to Beck, and brings us closer toward the bound conjectured by Burr and Erd}os. Applying the probabilistic method we also show that for all 1 and n + 1 there exists a bipartite graph with n vertices and maximum degree at most whose ramsey number is greater than c n for some absolute constant c > 1.

We consider bipartite subgraphs of sparse random graphs that are regular in the sense of Szemeredi and, among other things, show that they must satisfy a certain local pseudorandom property. This property and its consequences turn out to be useful when considering embedding problems in subgraphs of sparse random graphs.

For any positive integers r and n, let H(r, n) denote the family of graphs on n vertices with maximum degree r, and let H(r, n, n) denote the family of bipartite graphs H on 2n vertices with n vertices in each vertex class, and with maximum degree r. On one hand, we note that any H(r, n)-universal graph must have Ω(n2−2/r) edges. On the other hand, for any n ≥ n0(r), we explicitly construct H(r, n)-universal graphs G and Λ on n and 2n vertices, and with O(n 2−Ω ( 1 r log r) ) and O(n 2 − 1 r log 1/r n) edges, re-spectively, such that we can efficiently find a copy of any H ∈ H(r, n) in G deterministically. We also achieve sparse universal graphs using random constructions. Finally, we show that the bipartite random graph G = G(n, n, p), with 1 − p = cn 2r log 1/2r n is fault-tolerant; for a large enough constant c, even after deleting any α-fraction of the edges of G, the resulting graph is still H(r, a(α)n, a(α)n)-universal for some a: [0, 1) → (0, 1].

What is the minimum possible number of edges in a graph that contains a copy of every graph on n vertices with maximum degree a most k? This question, as well as several related variants, received a considerable amount of attention during the last decade. In this short survey we describe the known results focusing on the main ideas in the proofs, discuss the remaining open problems, and mention a recent application in the investigation of the complexity of subgraph containment problems.