### Abstract

The number T^{*}(n,k) is the least positive integer such that every graph with n = (_{2}^{k+1}) + t vertices (t ≥ 0) and at least T^{*}(n,k) edges contains k mutually vertex-disjoint complete subgraphs S_{1}, S_{2},..., S_{k} where S_{i} has i vertices, 1 ≤ i ≤ k. Obviously T^{*}(n, k) ≥ T(n, k), the Turán number of edges for a K_{k}. It is shown that if n ≥ 9 8k^{2} then equality holds and that there is ε{lunate} > 0 such that for (_{2}^{k+1}) ≤ n ≤ (_{2}^{k+1}) + ε{lunate}k^{2} inequality holds. Further T^{*}(n, k) is evaluated when k > k_{0}(t).

Original language | English |
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Pages (from-to) | 251-254 |

Number of pages | 4 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 23 |

Issue number | 2-3 |

DOIs | |

Publication status | Published - 1977 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Journal of Combinatorial Theory. Series B*,

*23*(2-3), 251-254. https://doi.org/10.1016/0095-8956(77)90038-7

**On a problem in extremal graph theory.** / Busolini, D. T.; Erdős, P.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory. Series B*, vol. 23, no. 2-3, pp. 251-254. https://doi.org/10.1016/0095-8956(77)90038-7

}

TY - JOUR

T1 - On a problem in extremal graph theory

AU - Busolini, D. T.

AU - Erdős, P.

PY - 1977

Y1 - 1977

N2 - The number T*(n,k) is the least positive integer such that every graph with n = (2k+1) + t vertices (t ≥ 0) and at least T*(n,k) edges contains k mutually vertex-disjoint complete subgraphs S1, S2,..., Sk where Si has i vertices, 1 ≤ i ≤ k. Obviously T*(n, k) ≥ T(n, k), the Turán number of edges for a Kk. It is shown that if n ≥ 9 8k2 then equality holds and that there is ε{lunate} > 0 such that for (2k+1) ≤ n ≤ (2k+1) + ε{lunate}k2 inequality holds. Further T*(n, k) is evaluated when k > k0(t).

AB - The number T*(n,k) is the least positive integer such that every graph with n = (2k+1) + t vertices (t ≥ 0) and at least T*(n,k) edges contains k mutually vertex-disjoint complete subgraphs S1, S2,..., Sk where Si has i vertices, 1 ≤ i ≤ k. Obviously T*(n, k) ≥ T(n, k), the Turán number of edges for a Kk. It is shown that if n ≥ 9 8k2 then equality holds and that there is ε{lunate} > 0 such that for (2k+1) ≤ n ≤ (2k+1) + ε{lunate}k2 inequality holds. Further T*(n, k) is evaluated when k > k0(t).

UR - http://www.scopus.com/inward/record.url?scp=49449127446&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=49449127446&partnerID=8YFLogxK

U2 - 10.1016/0095-8956(77)90038-7

DO - 10.1016/0095-8956(77)90038-7

M3 - Article

AN - SCOPUS:49449127446

VL - 23

SP - 251

EP - 254

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 2-3

ER -