{
  "id": "2004.07147",
  "version": "v1",
  "published": "2020-04-15T15:21:08.000Z",
  "updated": "2020-04-15T15:21:08.000Z",
  "title": "Turán and Ramsey-type results for unavoidable subgraphs",
  "authors": [
    "Alp Müyesser",
    "Michael Tait"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We study Tur\\'an and Ramsey-type problems on edge-colored graphs. An edge-colored graph is called {\\em $\\varepsilon$-balanced} if each color class contains at least an $\\varepsilon$-proportion of its edges. Given a family $\\mathcal{F}$ of edge-colored graphs, the Ramsey function $R(\\varepsilon, \\mathcal{F})$ is the smallest $n$ for which any $\\varepsilon$-balanced $K_n$ must contain a copy of an $F\\in\\mathcal{F}$, and the Tur\\'an function $\\mathrm{ex}(\\varepsilon, n, \\mathcal{F})$ is the maximum number of edges in an $n$-vertex $\\varepsilon$-balanced graph which avoids all of $\\mathcal{F}$. In this paper, we consider this Tur\\'an function for several classes of edge-colored graphs, we show that the Ramsey function is linear for bounded degree graphs, and we prove a theorem that gives a relationship between the two parameters.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-15T15:21:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ramsey-type results",
      "edge-colored graph",
      "unavoidable subgraphs",
      "ramsey function",
      "turan function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}