Answering a question raised by Dudek and Pra lat [4], we show that if pn → ∞, w.h.p., whenever G = G(n, p) is 2-coloured, there exists a monochromatic path of length n(2/3 + o(1)). This result is optimal in the sense that 2/3 cannot be replaced by a larger constant.As part of the proof we obtain the following result which may be of independent interest. We show that given a graph G on n vertices with at least (1 − ε) n 2 edges, whenever G is 2-edge-coloured, there is a monochromatic path of length at least (2/3 − 100 √ ε)n. This is an extension of the classical result by Gerencsér and Gyárfás [6] which says that whenever Kn is 2-coloured there is a monochromatic path of length at least 2n/3.

We study the problem of finding monotone subsequences in an array from the viewpoint of sublinear algorithms. For fixed k ∈ N and ε > 0, we show that the non-adaptive query complexity of finding a length-k monotone subsequence of f : [n] → R, assuming that f is ε-far from free of such subsequences, is Θ((log n) log 2 k ). Prior to our work, the best algorithm for this problem, due to Newman, Rabinovich, Rajendraprasad, and Sohler (2017), made (log n) O(k 2 ) non-adaptive queries; and the only lower bound known, of Ω(log n) queries for the case k = 2, followed from that on testing monotonicity due to Ergün, Kannan, Kumar, Rubinfeld, and Viswanathan (2000) and Fischer (2004).

The k‐color bipartite Ramsey number of a bipartite graph H is the least integer n for which every k‐edge‐colored complete bipartite graph Kn,n contains a monochromatic copy of H. The study of bipartite Ramsey numbers was initiated, over 40 years ago, by Faudree and Schelp and, independently, by Gyárfás and Lehel, who determined the 2‐color Ramsey number of paths. In this paper we determine asymptotically the 3‐color bipartite Ramsey number of paths and (even) cycles.

The k-colour bipartite Ramsey number of a bipartite graph H is the least integer N for which every k-edge-coloured complete bipartite graph K N,N contains a monochromatic copy of H. The study of bipartite Ramsey numbers was initiated over 40 years ago by Faudree and Schelp and, independently, by Gyárfás and Lehel, who determined the 2-colour bipartite Ramsey number of paths. Recently the 3-colour Ramsey number of paths and (even) cycles, was essentially determined as well. Improving the results of DeBiasio, Gyárfás, Krueger, Ruszinkó, and Sárközy, in this paper we determine asymptotically the 4-colour bipartite Ramsey number of paths and cycles. We also provide new upper bounds on the k-colour bipartite Ramsey numbers of paths and cycles which are close to being tight.Over the years, many generalisations of Ramsey numbers have been considered (an excellent survey [3] by Conlon, Fox and Sudakov contains many examples); one natural example that we consider here

The problem of finding dense induced bipartite subgraphs in H-free graphs has a long history, and was posed 30 years ago by Erdős, Faudree, Pach and Spencer. In this paper, we obtain several results in this direction. First we prove that any H-free graph with minimum degree at least d contains an induced bipartite subgraph of minimum degree at least c H log d/ log log d, thus nearly confirming one and proving another conjecture of Esperet, Kang and Thomassé. Complementing this result, we further obtain optimal bounds for this problem in the case of dense triangle-free graphs, and we also answer a question of Erdős, Janson,