### Abstract

This paper is a continuation of [10], where P. Erdo{combining double acute accent}s, A. Hajnal, V. T. Sós, and E. Szemerédi investigated the following problem: Assume that a so called forbidden graph L and a function f(n)=o(n) are fixed. What is the maximum number of edges a graph G _{n} on n vertices can have without containing L as a subgraph, and also without having more than f(n) independent vertices? This problem is motivated by the classical Turán and Ramsey theorems, and also by some applications of the Turán theorem to geometry, analysis (in particular, potential theory) [27-29], [11-13]. In this paper we are primarily interested in the following problem. Let (G _{n} ) be a graph sequence where G _{n} has n vertices and the edges of G _{n} are coloured by the colours χ _{1} ,...,χ _{r} so that the subgraph of colour χ _{υ} contains no complete subgraph K _{pv} , (v=1,... r). Further, assume that the size of any independent set in G _{n} is o(n) (as n→∞). What is the maximum number of edges in G _{n} under these conditions? One of the main results of this paper is the description of a procedure yielding relatively simple sequences of asymptotically extremal graphs for the problem. In a continuation of this paper we shall investigate the problem where instead of α(G _{n} )=o(n) we assume the stronger condition that the maximum size of a K _{p} -free induced subgraph of G _{n} is o(n).

Original language | English |
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Pages (from-to) | 31-56 |

Number of pages | 26 |

Journal | Combinatorica |

Volume | 13 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 1993 |

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### Keywords

- AMS subject classification code (1991): 05C35, 05C55

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Mathematics

### Cite this

*Combinatorica*,

*13*(1), 31-56. https://doi.org/10.1007/BF01202788