{
  "id": "2204.06874",
  "version": "v1",
  "published": "2022-04-14T10:54:27.000Z",
  "updated": "2022-04-14T10:54:27.000Z",
  "title": "A Note on Edge Colorings and Trees",
  "authors": [
    "Adi Jarden",
    "Ziv Shami"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We point out some connections between existence of homogenous sets for certain edge colorings and existence of branches in certain trees. As a consequence, we get that any locally additive coloring (a notion introduced in the paper) of a cardinal $\\kappa$ has a homogeneous set of size $\\kappa$ provided that the number of colors, $\\mu$ satisfies $\\mu^+<\\kappa$. Another result is that an uncountable cardinal $\\kappa$ is weakly compact if and only if $\\kappa$ is regular, has the tree property and for each $\\lambda,\\mu<\\kappa$ there exists $\\kappa^*<\\kappa$ such that every tree of height $\\mu$ with $\\lambda$ nodes has less than $\\kappa^*$ branches.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-14T10:54:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E02",
      "03E55"
    ],
    "keywords": [
      "edge colorings",
      "tree property",
      "uncountable cardinal",
      "consequence",
      "homogeneous set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}