Abstract
Building on classical theorems of Sperner and Kruskal-Katona, we investigate antichains \(\mathcal {F}\) in the Boolean lattice Bn of all subsets of \([n]:=\{1,2,\dots ,n\}\), where \(\mathcal {F}\) is flat, meaning that it contains sets of at most two consecutive sizes, say \(\mathcal {F}=\mathcal {A}\cup {\mathscr{B}}\), where \(\mathcal {A}\) contains only k-subsets, while \({\mathscr{B}}\) contains only (k − 1)-subsets. Moreover, we assume \(\mathcal {A}\) consists of the first m k-subsets in squashed (colexicographic) order, while \({\mathscr{B}}\) consists of all (k − 1)-subsets not contained in the subsets in \(\mathcal {A}\). Given reals α, β > 0, we say the weight of \(\mathcal {F}\) is \(\alpha \cdot |\mathcal {A}|+\beta \cdot |{\mathscr{B}}|\). We characterize the minimum weight antichains \(\mathcal {F}\) for any given n,k,α,β, and we do the same when in addition \(\mathcal {F}\) is a maximal antichain. We can then derive asymptotic results on both the minimum size and the minimum Lubell function.
Similar content being viewed by others
References
Anderson, I.: Combinatorics of finite sets. Courier Dover Publications (2002)
Bollobás, B.: On generalized graphs. Acta. Math. Acad. Sci. Hung. 16, 447–452 (1965)
Clements, G.F.: The minimal number of basic elements in a multiset antichain. J. Combin. Theory Ser. A 25, 153–162 (1978)
Engel, K.: Sperner Theory. Cambridge University Press (1997)
Frankl, P., Matsumoto, M., Ruzsa, I. Z., Tokushige, N.: Minimum shadows in uniform hypergraphs and a generalization of the Takagi function. J. Combinatorial Th. (ser. A) 69, 125–148 (1995)
Grüttmüller, M., Hartmann, S., Kalinowski, T., Leck, U., Roberts, I.T.: Maximal flat antichains of minimum weight. Electron. J. Combin. 16(1), #R69 (2009)
Griggs, J.R., Li, W.-T., Lu, L.: Diamond-free families. J. Combinatorial Theory Ser. A 119, 310–322 (2012)
Griggs, J.R., Li, W.-T.: Progress on poset-free families of subsets, recent trends in combinatorics. In: Beveridge, A., Griggs, J.R., Hogben, L., Musiker, G., Tetali, P. (eds.) The IMA volumes in mathematics and its applications, vol. 159, pp 317–338. Springer, Berlin (2016)
Kalinowski, T., Leck, U., Roberts, I.T.: Maximal antichains of minimum size. Electron. J. Combin. 20(2), #P3 (2013)
Katona, G.O.H.: A theorem of finite sets. In: Erdös, P., Katona, G. (eds.) Theory of graphs, pp 187–207. Akadémiai Kiadó and Academic Press, New York (1968)
Kisvölcsey, Á.: Flattening antichains. Combinatorica 308(11), 2247–2260 (2008)
Kruskal, J. B.: The optimal number of simplices in a complex. In: Bellman, R. (ed.) Mathemaical Optimization Techniques, pp 251–278. Univ. of California Press, Berkeley–Los Angeles (1963)
Lieby, P.: Extremal Problems in Finite Sets. PhD Thesis, Northern Territory University, Darwin (Australia) (1999)
Lieby, P.: Antichains on three levels. Electron. J. Combin. 11, #R50 (2004)
Lubell, D.: A short proof of Sperner’s lemma. J. Combin. Theory 1, 299 (1966)
Meshalkin, L.D.: Generalization of Sperner’s Theorem on the number of subsets of a finite set. Teor. Verojatnost. i Primenen 8, 219–220 (1963). English transl.: Theor. Probab. Appl. 8 (1963), 203–204
Sperner, E.: Ein Satz über Untermengen einer endlichen Menge. Math. Z 27, 544–548 (1928)
Topley, K.: Computationally efficient bounds for the sum of Catalan numbers. arXiv:1601.04223v2 [math.CO] (2016)
Yamamoto, K.: Logarithmic order of free distributive lattices. J. Math. Soc. Japan 6, 343–353 (1954)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Research supported in part by a grant from the Simons Foundation (#282896 to Jerrold Griggs) and by a long-term visiting position at the IMA, University of Minnesota.
Rights and permissions
About this article
Cite this article
Griggs, J.R., Hartmann, S., Kalinowski, T. et al. Minimum Weight Flat Antichains of Subsets. Order 38, 441–453 (2021). https://doi.org/10.1007/s11083-021-09550-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11083-021-09550-x