{
  "id": "1707.04297",
  "version": "v1",
  "published": "2017-07-13T20:14:06.000Z",
  "updated": "2017-07-13T20:14:06.000Z",
  "title": "The size-Ramsey number of powers of paths",
  "authors": [
    "Dennis Clemens",
    "Matthew Jenssen",
    "Yoshiharu Kohayakawa",
    "Natasha Morrison",
    "Guilherme Oliveira Mota",
    "Damian Reding",
    "Barnaby Roberts"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Given graphs $G$ and $H$ and a positive integer $q$ say that $G$ is $q$-Ramsey for $H$, denoted $G\\rightarrow (H)_q$, if every $q$-colouring of the edges of $G$ contains a monochromatic copy of $H$. The size-Ramsey number $\\hat{r}(H)$ of a graph $H$ is defined to be $\\hat{r}(H)=\\min\\{|E(G)|\\colon G\\rightarrow (H)_2\\}$. Answering a question of Conlon, we prove that, for every fixed $k$, we have $\\hat{r}(P_n^k)=O(n)$, where $P_n^k$ is the $k$-th power of the $n$-vertex path $P_n$ (i.e. , the graph with vertex set $V(P_n)$ and all edges $\\{u,v\\}$ such that the distance between $u$ and $v$ in $P_n$ is at most $k$). Our proof is probabilistic, but can also be made constructive.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-13T20:14:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "size-ramsey number",
      "monochromatic copy",
      "vertex set",
      "vertex path",
      "th power"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}