### Abstract

As a generalization of the concept of Sidon sets, a set of real numbers is called a (4, 5)-set if every four-element subset determines at least five distinct differences. Let g(n) be the largest number such that any n-element (4,5)-set contains a g(n)-element Sidon set (i.e., a subset of g(n) elements with distinct differences). It is shown that ( 1 2 + ε{lunate}) n ≤ g(n) ≤ 3n 5 + 1, where ε{lunate} is a positive constant. The main result is the lower bound whose proof is based on a Turán-type theorem obtained for sparse 3-uniform hypergraphs associated with (4, 5)-sets.

Original language | English |
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Pages (from-to) | 108-118 |

Number of pages | 11 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 64 |

Issue number | 1 |

DOIs | |

Publication status | Published - May 1995 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

### Cite this

*Journal of Combinatorial Theory. Series B*,

*64*(1), 108-118. https://doi.org/10.1006/jctb.1995.1028

**Linear Sets with Five Distinct Differences among Any Four Elements.** / Gyárfás, A.; Lehel, J.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory. Series B*, vol. 64, no. 1, pp. 108-118. https://doi.org/10.1006/jctb.1995.1028

}

TY - JOUR

T1 - Linear Sets with Five Distinct Differences among Any Four Elements

AU - Gyárfás, A.

AU - Lehel, J.

PY - 1995/5

Y1 - 1995/5

N2 - As a generalization of the concept of Sidon sets, a set of real numbers is called a (4, 5)-set if every four-element subset determines at least five distinct differences. Let g(n) be the largest number such that any n-element (4,5)-set contains a g(n)-element Sidon set (i.e., a subset of g(n) elements with distinct differences). It is shown that ( 1 2 + ε{lunate}) n ≤ g(n) ≤ 3n 5 + 1, where ε{lunate} is a positive constant. The main result is the lower bound whose proof is based on a Turán-type theorem obtained for sparse 3-uniform hypergraphs associated with (4, 5)-sets.

AB - As a generalization of the concept of Sidon sets, a set of real numbers is called a (4, 5)-set if every four-element subset determines at least five distinct differences. Let g(n) be the largest number such that any n-element (4,5)-set contains a g(n)-element Sidon set (i.e., a subset of g(n) elements with distinct differences). It is shown that ( 1 2 + ε{lunate}) n ≤ g(n) ≤ 3n 5 + 1, where ε{lunate} is a positive constant. The main result is the lower bound whose proof is based on a Turán-type theorem obtained for sparse 3-uniform hypergraphs associated with (4, 5)-sets.

UR - http://www.scopus.com/inward/record.url?scp=25744445650&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=25744445650&partnerID=8YFLogxK

U2 - 10.1006/jctb.1995.1028

DO - 10.1006/jctb.1995.1028

M3 - Article

AN - SCOPUS:25744445650

VL - 64

SP - 108

EP - 118

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 1

ER -