Directed Ramsey number for trees

In this paper, we study Ramsey-type problems for directed graphs. We first consider the $k$-colour oriented Ramsey number of $H$, denoted by $\overrightarrow{R}(H,k)$, which is the least $n$ for which every $k$-edge-coloured tournament on $n$ vertices contains a monochromatic copy of $H$. We prove that $\overrightarrow{R}(T,k) \le c_k|T|^k$ for any oriented tree $T$. This is a generalisation of a similar result for directed paths by Chv\'atal and by Gy\'arf\'as and Lehel, and answers a question of Yuster. In general, it is tight up to a constant factor. We also consider the $k$-colour directed Ramsey number $\overleftrightarrow{R}(H,k)$ of $H$, which is defined as above, but, instead of colouring tournaments, we colour the complete directed graph of order $n$. Here we show that $\overleftrightarrow{R}(T,k) \le c_k|T|^{k-1}$ for any oriented tree $T$, which is again tight up to a constant factor, and it generalises a result by Williamson and by Gy\'arf\'as and Lehel who determined the $2$-colour directed Ramsey number of directed paths.Comment: 27 pages, 14 figure

Directed Ramsey number for trees

We call a family F of subsets of [n] s-saturated if it contains no s pairwise disjoint sets, and moreover no set can be added to F while preserving this property (here [n] = {1, . . . , n}). More than 40 years ago, Erd˝os and Kleitman conjectured that an s-saturated family of subsets of [n] has size at least (1 − 2 −(s−1))2n. It is easy to show that every s-saturated family has size at least 1 2 · 2 n, but, as was mentioned by Frankl and Tokushige, even obtaining a slightly better bound of (1/2 + ε)2n, for some fixed ε > 0, seems difficult. In this note, we prove such a result, showing that every s-saturated family of subsets of [n] has size at least (1 − 1/s)2n. This lower bound is a consequence of a multipartite version of the problem, in which we seek a lower bound on |F1| + . . . + |Fs| where F1, . . . , Fs are families of subsets of [n], such that there are no s pairwise disjoint sets, one from each family Fi , and furthermore no set can be added to any of the families while preserving this property. We show that |F1| + . . . + |Fs| ≥ (s − 1) · 2 n, which is tight e.g. by taking F1 to be empty, and letting the remaining families be the families of all subsets of [n]

Constrained Ramsey Numbers

For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum n such that every edge coloring of the complete graph on n vertices, with any number of colors, has a monochromatic subgraph isomorphic to S or a rainbow (all edges differently colored) subgraph isomorphic to T. The Erdos-Rado Canonical Ramsey Theorem implies that f(S, T) exists if and only if S is a star or T is acyclic, and much work has been done to determine the rate of growth of f(S, T) for various types of parameters. When S and T are both trees having s and t edges respectively, Jamison, Jiang, and Ling showed that f(S, T) <= O(st^2) and conjectured that it is always at most O(st). They also mentioned that one of the most interesting open special cases is when T is a path. In this work, we study this case and show that f(S, P_t) = O(st log t), which differs only by a logarithmic factor from the conjecture. This substantially improves the previous bounds for most values of s and t.Comment: 12 pages; minor revision

Embedding large subgraphs into dense graphs

What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac's theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect matchings are generalized by perfect F-packings, where instead of covering all the vertices of G by disjoint edges, we want to cover G by disjoint copies of a (small) graph F. It is unlikely that there is a characterization of all graphs G which contain a perfect F-packing, so as in the case of Dirac's theorem it makes sense to study conditions on the minimum degree of G which guarantee a perfect F-packing. The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy and Szemeredi have proved to be powerful tools in attacking such problems and quite recently, several long-standing problems and conjectures in the area have been solved using these. In this survey, we give an outline of recent progress (with our main emphasis on F-packings, Hamiltonicity problems and tree embeddings) and describe some of the methods involved

On Ramsey properties of classes with forbidden trees

Let F be a set of relational trees and let Forbh(F) be the class of all structures that admit no homomorphism from any tree in F; all this happens over a fixed finite relational signature $\sigma$. There is a natural way to expand Forbh(F) by unary relations to an amalgamation class. This expanded class, enhanced with a linear ordering, has the Ramsey property.Comment: Keywords: forbidden substructure; amalgamation; Ramsey class; partite method v2: changed definition of expanded class; v3: final versio