{
  "id": "1808.09863",
  "version": "v1",
  "published": "2018-08-29T14:58:08.000Z",
  "updated": "2018-08-29T14:58:08.000Z",
  "title": "Ramsey numbers of Berge-hypergraphs and related structures",
  "authors": [
    "Nika Salia",
    "Casey Tompkins",
    "Zhiyu Wang",
    "Oscar Zamora"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "For a graph $G=(V,E)$, a hypergraph $\\mathcal{H}$ is called a Berge-$G$, denoted by $BG$, if there exists a bijection $f: E(G) \\to E(\\mathcal{H})$ such that for every $e \\in E(G)$, $e \\subseteq f(e)$. Let the Ramsey number $R^r(BG,BG)$ be the smallest integer $n$ such that for any $2$-edge-coloring of a complete $r$-uniform hypergraph on $n$ vertices, there is a monochromatic Berge-$G$ subhypergraph. In this paper, we show that the 2-color Ramsey number of Berge cliques is linear. In particular, we show that $R^3(BK_s, BK_t) = s+t-3$ for $s,t \\geq 4$ and $\\max(s,t) \\geq 5$ where $BK_n$ is a Berge-$K_n$ hypergraph. For higher uniformity, we show that $R^4(BK_t, BK_t) = t+1$ for $t\\geq 6$ and $R^k(BK_t, BK_t)=t$ for $k \\geq 5$ and $t$ sufficiently large. We also investigate the Ramsey number of trace hypergraphs, suspension hypergraphs and expansion hypergraphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-29T14:58:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ramsey number",
      "related structures",
      "berge-hypergraphs",
      "suspension hypergraphs",
      "expansion hypergraphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}