{
  "id": "2409.06650",
  "version": "v1",
  "published": "2024-09-10T17:13:38.000Z",
  "updated": "2024-09-10T17:13:38.000Z",
  "title": "Induced subgraphs of $K_r$-free graphs and the Erdős--Rogers problem",
  "authors": [
    "Lior Gishboliner",
    "Oliver Janzer",
    "Benny Sudakov"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "For two graphs $F,H$ and a positive integer $n$, the function $f_{F,H}(n)$ denotes the largest $m$ such that every $H$-free graph on $n$ vertices contains an $F$-free induced subgraph on $m$ vertices. This function has been extensively studied in the last 60 years when $F$ and $H$ are cliques and became known as the Erd\\H{o}s-Rogers function. Recently, Balogh, Chen and Luo, and Mubayi and Verstra\\\"ete initiated the systematic study of this function in the case where $F$ is a general graph. Answering, in a strong form, a question of Mubayi and Verstra\\\"ete, we prove that for every positive integer $r$ and every $K_{r-1}$-free graph $F$, there exists some $\\varepsilon_F>0$ such that $f_{F,K_r}(n)=O(n^{1/2-\\varepsilon_F})$. This result is tight in two ways. Firstly, it is no longer true if $F$ contains $K_{r-1}$ as a subgraph. Secondly, we show that for all $r\\geq 4$ and $\\varepsilon>0$, there exists a $K_{r-1}$-free graph $F$ for which $f_{F,K_r}(n)=\\Omega(n^{1/2-\\varepsilon})$. Along the way of proving this, we show in particular that for every graph $F$ with minimum degree $t$, we have $f_{F,K_4}(n)=\\Omega(n^{1/2-6/\\sqrt{t}})$. This answers (in a strong form) another question of Mubayi and Verstra\\\"ete. Finally, we prove that there exist absolute constants $0<c<C$ such that for each $r\\geq 4$, if $F$ is a bipartite graph with sufficiently large minimum degree, then $\\Omega(n^{\\frac{c}{\\log r}})\\leq f_{F,K_r}(n)\\leq O(n^{\\frac{C}{\\log r}})$. This shows that for graphs $F$ with large minimum degree, the behaviour of $f_{F,K_r}(n)$ is drastically different from that of the corresponding off-diagonal Ramsey number $f_{K_2,K_r}(n)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-10T17:13:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "free graph",
      "induced subgraph",
      "erdős-rogers problem",
      "strong form",
      "positive integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}