### Abstract

Let the K_{p}-independence number α_{p} (G) of a graph G be the maximum order of an induced subgraph in G that contains no K_{p}. (So K_{2}-independence number is just the maximum size of an independent set.) For given integers r, p, m > 0 and graphs L_{1},…,L_{r}, we define the corresponding Turán-Ramsey function RT_{p}(n, L_{1},…,L_{r}, m) to be the maximum number of edges in a graph G_{n} of order n such that α_{p}(G_{n}) ≤ m and there is an edge-colouring of G with r colours such that the j^{th} colour class contains no copy of L_{j}, for j = 1,…, r. In this continuation of [11] and [12], we will investigate the problem where, instead of α(G_{n}) = o(n), we assume (for some fixed p > 2) the stronger condition that α_{p}(G_{n}) = o(n). The first part of the paper contains multicoloured Turán-Ramsey theorems for graphs G_{n} of order n with small K_{p}-independence number α_{p}(G_{n}). Some structure theorems are given for the case α_{p}(G_{n}) = o(n), showing that there are graphs with fairly simple structure that are within o(n^{2}) of the extremal size; the structure is described in terms of the edge densities between certain sets of vertices. The second part of the paper is devoted to the case r = 1, i.e., to the problem of determining the asymptotic value of [formula omitted] for p <q. Several results are proved, and some other problems and conjectures are stated.

Original language | English |
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Pages (from-to) | 297-325 |

Number of pages | 29 |

Journal | Combinatorics Probability and Computing |

Volume | 3 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1994 |

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### ASJC Scopus subject areas

- Applied Mathematics
- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics

### Cite this

_{p}-Independence Numbers.

*Combinatorics Probability and Computing*,

*3*(3), 297-325. https://doi.org/10.1017/S0963548300001218