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
Alon, Bohman, Holzman and Kleitman proved that any partition of a d-dimensional discrete box into proper sub-boxes must consist of at least 2 d sub-boxes. Recently, Leader, Milićević and Tan considered the question of how many odd-sized proper boxes are needed to partition a d-dimensional box of odd size, and they asked whether the trivial construction consisting of 3 d boxes is best possible. We show that approximately 2.93 d boxes are enough, and consider some natural generalisations. Question 1.2 (Leader, Milićević, Tan [7]). Let A be a d-dimensional odd box, and let {B 1 , B 2 , . . . , B m } be a partition of A into odd proper sub-boxes. Does it follow that then m ≥ 3 d ?.Our first result is that the answer to this question is 'no': Theorem 1.3. Let d ∈ Z + be divisible by 3. Then there exists a partition of [5] d into 25 d/3 ≤ 2.93 d odd proper sub-boxes.
A well-known theorem of Chung and Graham states that if h ≥ 4 then a tournament T is quasirandom if and only if T contains each h-vertex tournament the 'correct number' of times as a subtournament. In this paper we investigate the relationship between quasirandomness of T and the count of a single h-vertex tournament H in T . We consider two types of counts, the global one and the local one.We first observe that if T has the correct global count of H and h ≥ 7 then quasirandomness of T is only forced if H is transitive. The next natural question when studying quasirandom objects asks whether possessing the correct local counts of H is enough to force quasirandomness of T . A tournament H is said to be locally forcing if it has this property.Variants of the local forcing problem have been studied before in both the graph and hypergraph settings. Perhaps the closest analogue of our problem was considered by Simonovits and Sós who looked at whether having 'correct counts' of a fixed graph H as an induced subgraph of G implies G must be quasirandom, in an appropriate sense. They proved that this is indeed the case when H is regular and conjectured that it holds for all H (except the path on 3 vertices). Contrary to the Simonovits-Sós conjecture, in the tournament setting we prove that a constant proportion of all tournaments are not locally forcing. In fact, any locally forcing tournament must itself be strongly quasirandom. On the other hand, unlike the global forcing case, we construct infinite families of non-transitive locally forcing tournaments.
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