### 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 |
---|---|

Pages (from-to) | 31-56 |

Number of pages | 26 |

Journal | Combinatorica |

Volume | 13 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 1993 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Mathematics(all)

### Cite this

*Combinatorica*,

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

**Turán-Ramsey theorems and simple asymptotically extremal structures.** / Erdős, P.; Hajnal, A.; Simonovits, M.; Sós, V. T.; Szemerédi, E.

Research output: Contribution to journal › Article

*Combinatorica*, vol. 13, no. 1, pp. 31-56. https://doi.org/10.1007/BF01202788

}

TY - JOUR

T1 - Turán-Ramsey theorems and simple asymptotically extremal structures

AU - Erdős, P.

AU - Hajnal, A.

AU - Simonovits, M.

AU - Sós, V. T.

AU - Szemerédi, E.

PY - 1993/3

Y1 - 1993/3

N2 - 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 Gn 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 (Gn) be a graph sequence where Gn has n vertices and the edges of Gn are coloured by the colours χ1,...,χr so that the subgraph of colour χυ contains no complete subgraph Kpv, (v=1,... r). Further, assume that the size of any independent set in Gn is o(n) (as n→∞). What is the maximum number of edges in Gn 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(Gn)=o(n) we assume the stronger condition that the maximum size of a Kp-free induced subgraph of Gn is o(n).

AB - 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 Gn 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 (Gn) be a graph sequence where Gn has n vertices and the edges of Gn are coloured by the colours χ1,...,χr so that the subgraph of colour χυ contains no complete subgraph Kpv, (v=1,... r). Further, assume that the size of any independent set in Gn is o(n) (as n→∞). What is the maximum number of edges in Gn 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(Gn)=o(n) we assume the stronger condition that the maximum size of a Kp-free induced subgraph of Gn is o(n).

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

UR - http://www.scopus.com/inward/record.url?scp=0038024953&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0038024953&partnerID=8YFLogxK

U2 - 10.1007/BF01202788

DO - 10.1007/BF01202788

M3 - Article

VL - 13

SP - 31

EP - 56

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 1

ER -