An r-edge coloring of a graph or hypergraph G " pV, Eq is a map c : E Ñ t0, . . . , r1u. Extending results of Rado and answering questions of Rado, Gyárfás and Sárközy we prove that• the vertex set of every r-edge colored countably infinite complete k-uniform hypergraph can be partitioned into r monochromatic tight paths with distinct colors (a tight path in a k-uniform hypergraph is a sequence of distinct vertices such that every set of k consecutive vertices forms an edge);• for all natural numbers r and k there is a natural number M such that the vertex set of every r-edge colored countably infinite complete graph can be partitioned into M monochromatic k th powers of paths apart from a finite set

Motivated by an old conjecture of P. Erdős and V. Neumann-Lara, our aim is to investigate digraphs with uncountable dichromatic number and orientations of undirected graphs with uncountable chromatic number. A graph has uncountable chromatic number if its vertices cannot be covered by countably many independent sets, and a digraph has uncountable dichromatic number if its vertices cannot be covered by countably many acyclic sets. We prove that, consistently, there are digraphs with uncountable dichromatic number and arbitrarily large digirth; this is in surprising contrast with the undirected case: any graph with uncountable chromatic number contains a 4-cycle. Next, we prove that several well-known graphs (uncountable complete graphs, certain comparability graphs, and shift graphs) admit orientations with uncountable dichromatic number in ZFC. However, we show that the statement "every graph of size and chromatic number 1 has an orientation with uncountable dichromatic number" is independent of ZFC. We end the article with several open problems. K E Y W O R D Sacyclic, chromatic number, dichromatic number, digraph, girth, orientation, partition INTRODUCTIONThe chromatic number of an undirected graph , denoted by ( ), is the minimal number of independent sets needed to cover the vertex set of . A beautiful branch of graph theory deals with the 606

We explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already, we significantly broaden this framework by developing the corresponding technique for countably closed models of size continuum. The applications range from various theorems on paradoxical decompositions of the plane, to coloring sparse set systems, results on graph chromatic number and constructions from point-set topology. Our main purpose is to demonstrate the ease and wide applicability of this method in a form accessible to anyone with a basic background in set theory and logic.

Our goal is to study the pseudo-intersection and tower numbers on uncountable regular cardinals, whether these two cardinal characteristics are necessarily equal, and related problems on the existence of gaps. First, we prove that either p(κ) = t(κ) or there is a (p(κ), λ)-gap of club-supported slaloms for some λ < p(κ). While the existence of such gaps is unclear, this is a promising step to lift Malliaris and Shelah's proof of p = t to uncountable cardinals. We do analyze gaps of slaloms and, in particular, show that p(κ) is always regular; the latter extends results of Garti. Finally, we turn to club variants of p(κ) and present a new model for the inequality p(κ) = κ + < p cl (κ) = 2 κ . In contrast to earlier arguments by Shelah and Spasojević, we achieve this by adding κ-Cohen reals and then successively diagonalizing the club filter; the latter is shown to preserve a Cohen witness to p(κ) = κ + .

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