×

An asymptotic version of Frankl’s conjecture. (English) Zbl 1470.05159

Summary: We show that the maximum average set size in complements of union-closed families over \(n\) elements is \((n+1)/2\). This implies an asymptotic version of Frankl’s conjecture.

MSC:

05D05 Extremal set theory
05A20 Combinatorial inequalities
05C35 Extremal problems in graph theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Balla, I.; Bollobás, B.; Eccles, T., On union-closed families of sets, J.Combin.Theory (Series A), 120, 3, 531-544 (2013) · Zbl 1259.05177
[2] Bruhn, H.; Charbit, P.; Schaudt., O.; Telle, J. A., The graph formulation of the union-closed sets conjecture, European J. Combin., 43, 210-219 (2015) · Zbl 1301.05183
[3] Bruhn, H.; Schaudt, O., The journey of the union-closed sets conjecture, Graphs Combin, 31, 6, 2043-2074 (2015) · Zbl 1327.05249
[4] Karpas, I., Two results on union-closed families, arxiv.org/abs/1708.01434 (2017)
[5] Maßberg, J., The union-closed sets conjecture for small families, Graphs Combin., 32, 5, 2047-2051 (2016) · Zbl 1440.05192
[6] Polymath 11. (2016). Gowers’s blog. gowers.wordpress.com/2016/01/21/frankls-union-closed- conjecture-a-possible-polymath-project/
[7] Poonen, B., Union-closed families, J.Combin.Theory (Series A), 59, 2, 253-268 (1992) · Zbl 0758.05096 · doi:10.1016/0097-3165(92)90068-6
[8] Reimer, D., An average set size theorem, Combin. Probab. Comput, 12, 1, 89-93 (2003) · Zbl 1013.05083 · doi:10.1017/S0963548302005230
[9] Rival, I., NATO ASI Series, 147, Graphs and Order (1985), Dordrecht: D. Reidel, Dordrecht
[10] Salzborn, F. (1989). A note on the intersecting sets conjecture. manuscript.
[11] Vučković, B.; Živković, M., The 12-element case of Frankl’s conjecture, IPSI Trans. Internet Res, 13, 1, 65-71 (2017)
[12] Wójcik, P., Density of union-closed families, Discrete Math, 105, 1-3, 259-267 (1992) · Zbl 0763.04006 · doi:10.1016/0012-365X(92)90148-9
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.