# On the number of union-free families

- 66 Downloads

## Abstract

A family of sets is union-free if there are no three distinct sets in the family such that the union of two of the sets is equal to the third set. Kleitman proved that every union-free family has size at most (1+*o*(1))( _{ n/2} ^{ n } ). Later, Burosch–Demetrovics–Katona–Kleitman–Sapozhenko asked for the number *α*(*n*) of such families, and they proved that \({2^{\left( {\begin{array}{*{20}{c}} n \\ {n/2} \end{array}} \right)}} \leqslant \alpha \left( n \right) \leqslant {2^{2\sqrt 2 \left( {\begin{array}{*{20}{c}} n \\ {n/2} \end{array}} \right)\left( {1 + o\left( 1 \right)} \right)}}\) They conjectured that the constant \(2\sqrt 2 \) can be removed in the exponent of the right-hand side. We prove their conjecture by formulating a new container-type theorem for rooted hypergraphs.

## Preview

Unable to display preview. Download preview PDF.

### References

- [1]N. Alon and F. Chung,
*Explicit construction of linear sized tolerant networks*, Annals Discrete Mathematics**38**(1988), 15–19.MathSciNetCrossRefMATHGoogle Scholar - [2]J. Balogh, R. Morris and W. Samotij,
*Independent sets in hypergraphs*, Journal of the American Mathematical Society**28**(2015), 669–709.MathSciNetCrossRefMATHGoogle Scholar - [3]G. Burosch, J. Demetrovics, G. O. H. Katona, D. J. Kleitman and A. A. Sapozhenko,
*On the number of databases and closure operations*, Theoretical Computer Science**78**(1991), 377–381.MathSciNetCrossRefMATHGoogle Scholar - [4]D. Kleitman,
*Extremal properties of collections of subsets containing no two sets and their union*, Journal of Combinatorial Theory. Series A**20**(1976), 390–392.MathSciNetCrossRefMATHGoogle Scholar - [5]L. Lovász,
*On the Shannon capacity of a graph*, IEEE Transactions on Information Theory**25**(1979), 1–7.MathSciNetCrossRefMATHGoogle Scholar - [6]R. Morris and D. Saxton,
*The number of C2’-free graphs*, Advances in Mathematics**298**(2016), 534–580.MathSciNetCrossRefMATHGoogle Scholar - [7]M. Saks,
*Kleitman and combinatorics*, Discrete Mathematics**257**(2002), 225–247.MathSciNetCrossRefMATHGoogle Scholar - [8]D. Saxton and A. Thomason,
*Hypergraph containers*, Inventiones mathematicae**201**(2015), 925–992.MathSciNetCrossRefMATHGoogle Scholar - [9]S. Ulam,
*A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics*, Vol. 8, Interscience, New York–London, 1960.Google Scholar