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 n/2 ). 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.
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References
N. Alon and F. Chung, Explicit construction of linear sized tolerant networks, Annals Discrete Mathematics 38 (1988), 15–19.
J. Balogh, R. Morris and W. Samotij, Independent sets in hypergraphs, Journal of the American Mathematical Society 28 (2015), 669–709.
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.
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.
L. Lovász, On the Shannon capacity of a graph, IEEE Transactions on Information Theory 25 (1979), 1–7.
R. Morris and D. Saxton, The number of C2’-free graphs, Advances in Mathematics 298 (2016), 534–580.
M. Saks, Kleitman and combinatorics, Discrete Mathematics 257 (2002), 225–247.
D. Saxton and A. Thomason, Hypergraph containers, Inventiones mathematicae 201 (2015), 925–992.
S. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, Vol. 8, Interscience, New York–London, 1960.
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Research is partially supported by Simons Fellowship, NSA Grant H98230-15-1-0002, NSF Grant DMS-1500121 and Arnold O. Beckman Research Award (UIUC Campus Research Board 15006).
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Balogh, J., Wagner, A.Z. On the number of union-free families. Isr. J. Math. 219, 431–448 (2017). https://doi.org/10.1007/s11856-017-1486-y
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DOI: https://doi.org/10.1007/s11856-017-1486-y