818 research outputs found

    Davies-trees in infinite combinatorics

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    This short note, prepared for the Logic Colloquium 2014, provides an introduction to Davies-trees and presents new applications in infinite combinatorics. In particular, we give new and simple proofs to the following theorems of P. Komj\'ath: every nn-almost disjoint family of sets is essentially disjoint for any n∈Nn\in \mathbb N; R2\mathbb R^2 is the union of n+2n+2 clouds if the continuum is at most ℵn\aleph_n for any n∈Nn\in \mathbb N; every uncountably chromatic graph contains nn-connected uncountably chromatic subgraphs for every n∈Nn\in \mathbb N.Comment: 8 pages, prepared for the Logic Colloquium 201

    Decompositions of edge-colored infinite complete graphs into monochromatic paths

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    An rr-edge coloring of a graph or hypergraph G=(V,E)G=(V,E) is a map c:E→{0,…,r−1}c:E\to \{0, \dots, r-1\}. Extending results of Rado and answering questions of Rado, Gy\'arf\'as and S\'ark\"ozy we prove that (1.) the vertex set of every rr-edge colored countably infinite complete kk-uniform hypergraph can be partitioned into rr monochromatic tight paths with distinct colors (a tight path in a kk-uniform hypergraph is a sequence of distinct vertices such that every set of kk consecutive vertices forms an edge), (2.) for all natural numbers rr and kk there is a natural number MM such that the vertex set of every rr-edge colored countably infinite complete graph can be partitioned into MM monochromatic kthk^{th} powers of paths apart from a finite set (a kthk^{th} power of a path is a sequence v0,v1,…v_0, v_1, \dots of distinct vertices such that 1≤∣i−j∣≤k1\le|i-j| \le k implies that vivjv_iv_j is an edge), (3.) the vertex set of every 22-edge colored countably infinite complete graph can be partitioned into 44 monochromatic squares of paths, but not necessarily into 33, (4.) the vertex set of every 22-edge colored complete graph on ω1\omega_1 can be partitioned into 22 monochromatic paths with distinct colors
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