### Abstract

For a property Γ and a family of sets F, let f(F,Γ) be the size of the largest subfamily of F having property Γ. For a positive integer m, let f(m, Γ) be the minimum of f(F,Γ) over all families of size m. A family F is said to be B_{d}-free if it has no subfamily F'={FI:I⊆[d]} of 2^{d} distinct sets such that for every I, J⊆[d], both F_{I}∪F_{J}=F_{I∪J} and F_{I}∩F_{J}=F_{I∩J} hold. A family F is a-union free if F_{1}∪⋯∪F_{a}≠F_{a+1} whenever F_{1},...,F_{a+1} are distinct sets in F. We verify a conjecture of Erdos and Shelah that f(m, B_{2}-free)=Θ(m^{2/3}). We also obtain lower and upper bounds for f(m, B_{d}-free) and f(m, a-union free).

Original language | English |
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Pages (from-to) | 101-104 |

Number of pages | 4 |

Journal | Electronic Notes in Discrete Mathematics |

Volume | 38 |

DOIs | |

Publication status | Published - Dec 1 2011 |

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### Keywords

- B-free subfamilies
- Extremal set theory
- Union-free subfamilies

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

_{d}-free and union-free subfamilies.

*Electronic Notes in Discrete Mathematics*,

*38*, 101-104. https://doi.org/10.1016/j.endm.2011.09.017

**Large B _{d}-free and union-free subfamilies.** / Barát, János; Füredi, Z.; Kantor, Ida; Kim, Younjin; Patkós, Balázs.

Research output: Contribution to journal › Article

_{d}-free and union-free subfamilies',

*Electronic Notes in Discrete Mathematics*, vol. 38, pp. 101-104. https://doi.org/10.1016/j.endm.2011.09.017

_{d}-free and union-free subfamilies. Electronic Notes in Discrete Mathematics. 2011 Dec 1;38:101-104. https://doi.org/10.1016/j.endm.2011.09.017

}

TY - JOUR

T1 - Large Bd-free and union-free subfamilies

AU - Barát, János

AU - Füredi, Z.

AU - Kantor, Ida

AU - Kim, Younjin

AU - Patkós, Balázs

PY - 2011/12/1

Y1 - 2011/12/1

N2 - For a property Γ and a family of sets F, let f(F,Γ) be the size of the largest subfamily of F having property Γ. For a positive integer m, let f(m, Γ) be the minimum of f(F,Γ) over all families of size m. A family F is said to be Bd-free if it has no subfamily F'={FI:I⊆[d]} of 2d distinct sets such that for every I, J⊆[d], both FI∪FJ=FI∪J and FI∩FJ=FI∩J hold. A family F is a-union free if F1∪⋯∪Fa≠Fa+1 whenever F1,...,Fa+1 are distinct sets in F. We verify a conjecture of Erdos and Shelah that f(m, B2-free)=Θ(m2/3). We also obtain lower and upper bounds for f(m, Bd-free) and f(m, a-union free).

AB - For a property Γ and a family of sets F, let f(F,Γ) be the size of the largest subfamily of F having property Γ. For a positive integer m, let f(m, Γ) be the minimum of f(F,Γ) over all families of size m. A family F is said to be Bd-free if it has no subfamily F'={FI:I⊆[d]} of 2d distinct sets such that for every I, J⊆[d], both FI∪FJ=FI∪J and FI∩FJ=FI∩J hold. A family F is a-union free if F1∪⋯∪Fa≠Fa+1 whenever F1,...,Fa+1 are distinct sets in F. We verify a conjecture of Erdos and Shelah that f(m, B2-free)=Θ(m2/3). We also obtain lower and upper bounds for f(m, Bd-free) and f(m, a-union free).

KW - B-free subfamilies

KW - Extremal set theory

KW - Union-free subfamilies

UR - http://www.scopus.com/inward/record.url?scp=82955245625&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=82955245625&partnerID=8YFLogxK

U2 - 10.1016/j.endm.2011.09.017

DO - 10.1016/j.endm.2011.09.017

M3 - Article

AN - SCOPUS:82955245625

VL - 38

SP - 101

EP - 104

JO - Electronic Notes in Discrete Mathematics

JF - Electronic Notes in Discrete Mathematics

SN - 1571-0653

ER -